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Proof step 5 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]]) ⊦ ∀ key. key ∈ keys(fs, ax0).toSet → key ∈ dom ri

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 6:
# ax0 = # ax1, key ∈ keys(fs, ax0), ax0.toSet ⊆ range ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]) ⊦ key ∈ ri

Rule argument

Additional information

Used simplifier rules: ⊦ key ∈ dom ri ↔ key ∈ ri; (s)% ⊦ key ∈ kx.toSet ↔ key ∈ kx; (s)