Theorem ids1 14533

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 6855	. . . . 5 ⊢ ( I ‘𝐴) ∈ V
2		fvi 6918	. . . . 5 ⊢ (( I ‘𝐴) ∈ V → ( I ‘( I ‘𝐴)) = ( I ‘𝐴))
3	1, 2	ax-mp 5	. . . 4 ⊢ ( I ‘( I ‘𝐴)) = ( I ‘𝐴)
4	3	opeq2i 4835	. . 3 ⊢ ⟨0, ( I ‘( I ‘𝐴))⟩ = ⟨0, ( I ‘𝐴)⟩
5	4	sneqi 4593	. 2 ⊢ {⟨0, ( I ‘( I ‘𝐴))⟩} = {⟨0, ( I ‘𝐴)⟩}
6		df-s1 14532	. 2 ⊢ ⟨“( I ‘𝐴)”⟩ = {⟨0, ( I ‘( I ‘𝐴))⟩}
7		df-s1 14532	. 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
8	5, 6, 7	3eqtr4ri 2771	1 ⊢ ⟨“𝐴”⟩ = ⟨“( I ‘𝐴)”⟩

Syntax hints:   = wceq 1542   ∈ wcel 2114  Vcvv 3442  {csn 4582  ⟨cop 4588   I cid 5526  ‘cfv 6500  0cc0 11038  ⟨“cs1 14531

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-s1 14532

This theorem is referenced by:  s1prc  14540  s1cli  14541  revs1  14700