Theorem ids1 14608

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

Assertion

Ref	Expression
ids1	⊢ ⟨“𝐴”⟩ = ⟨“( I ‘𝐴)”⟩

Proof of Theorem ids1

Step	Hyp	Ref	Expression
1		fvex 6876	. . . . 5 ⊢ ( I ‘𝐴) ∈ V
2		fvi 6939	. . . . 5 ⊢ (( I ‘𝐴) ∈ V → ( I ‘( I ‘𝐴)) = ( I ‘𝐴))
3	1, 2	ax-mp 5	. . . 4 ⊢ ( I ‘( I ‘𝐴)) = ( I ‘𝐴)
4	3	opeq2i 4834	. . 3 ⊢ ⟨0, ( I ‘( I ‘𝐴))⟩ = ⟨0, ( I ‘𝐴)⟩
5	4	sneqi 4592	. 2 ⊢ {⟨0, ( I ‘( I ‘𝐴))⟩} = {⟨0, ( I ‘𝐴)⟩}
6		df-s1 14607	. 2 ⊢ ⟨“( I ‘𝐴)”⟩ = {⟨0, ( I ‘( I ‘𝐴))⟩}
7		df-s1 14607	. 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
8	5, 6, 7	3eqtr4ri 2795	1 ⊢ ⟨“𝐴”⟩ = ⟨“( I ‘𝐴)”⟩

Syntax hints:   = wceq 1559   ∈ wcel 2141  Vcvv 3453  {csn 4581  ⟨cop 4587   I cid 5539  ‘cfv 6517  0cc0 11070  ⟨“cs1 14606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-s1 14607

This theorem is referenced by:  s1prc  14615  s1cli  14616  revs1  14775