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Proof step 3 from proof index-update-dom (previous step)

# ax0 = # ax1, ax0.toSet ⊆ range ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]) ⊦ keys(fs, ax0).toSet ⊆ dom ri

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ ks ⊆ ks0 ↔ (∀ key. key ∈ ks → key ∈ ks0)
Proof step 5:
# ax0 = # ax1, ax0.toSet ⊆ range ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]) ⊦ ∀ key. key ∈ keys(fs, ax0).toSet → key ∈ dom ri

Rule argument

spec: set-basic, inst: node-index name: Subset seq: ⊦ ks ⊆ ks0 ↔ (∀ key. key ∈ ks → key ∈ ks0) subst: [ks0, ks] → [dom ri, keys(fs, ax0).toSet] flags: <all paths>

Additional information

Used terms: dom ri, keys(fs, ax0).toSet