Theorem ids1 14543

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 6901	. . . . 5 ⊢ ( I ‘𝐴) ∈ V
2		fvi 6964	. . . . 5 ⊢ (( I ‘𝐴) ∈ V → ( I ‘( I ‘𝐴)) = ( I ‘𝐴))
3	1, 2	ax-mp 5	. . . 4 ⊢ ( I ‘( I ‘𝐴)) = ( I ‘𝐴)
4	3	opeq2i 4876	. . 3 ⊢ ⟨0, ( I ‘( I ‘𝐴))⟩ = ⟨0, ( I ‘𝐴)⟩
5	4	sneqi 4638	. 2 ⊢ {⟨0, ( I ‘( I ‘𝐴))⟩} = {⟨0, ( I ‘𝐴)⟩}
6		df-s1 14542	. 2 ⊢ ⟨“( I ‘𝐴)”⟩ = {⟨0, ( I ‘( I ‘𝐴))⟩}
7		df-s1 14542	. 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
8	5, 6, 7	3eqtr4ri 2771	1 ⊢ ⟨“𝐴”⟩ = ⟨“( I ‘𝐴)”⟩

Syntax hints:   = wceq 1541   ∈ wcel 2106  Vcvv 3474  {csn 4627  ⟨cop 4633   I cid 5572  ‘cfv 6540  0cc0 11106  ⟨“cs1 14541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-s1 14542

This theorem is referenced by:  s1prc  14550  s1cli  14551  revs1  14711