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Metamath Proof Explorer

Theorem ids1

Description: Identity function protection for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016)

Ref Expression
Assertion ids1 ⊢ ⟨“ A ”⟩ = ⟨“ I ⁡ A ”⟩

Proof

Step Hyp Ref Expression
1 fvex ⊢ I ⁡ A ∈ V
2 fvi ⊢ I ⁡ A ∈ V → I ⁡ I ⁡ A = I ⁡ A
3 1 2 ax-mp ⊢ I ⁡ I ⁡ A = I ⁡ A
4 3 opeq2i ⊢ 0 I ⁡ I ⁡ A = 0 I ⁡ A
5 4 sneqi ⊢ 0 I ⁡ I ⁡ A = 0 I ⁡ A
6 df-s1 ⊢ ⟨“ I ⁡ A ”⟩ = 0 I ⁡ I ⁡ A
7 df-s1 ⊢ ⟨“ A ”⟩ = 0 I ⁡ A
8 5 6 7 3eqtr4ri ⊢ ⟨“ A ”⟩ = ⟨“ I ⁡ A ”⟩