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Theorem ids1 14492
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
StepHypRef Expression
1 fvex 6860 . . . . 5 ( I ‘𝐴) ∈ V
2 fvi 6922 . . . . 5 (( I ‘𝐴) ∈ V → ( I ‘( I ‘𝐴)) = ( I ‘𝐴))
31, 2ax-mp 5 . . . 4 ( I ‘( I ‘𝐴)) = ( I ‘𝐴)
43opeq2i 4839 . . 3 ⟨0, ( I ‘( I ‘𝐴))⟩ = ⟨0, ( I ‘𝐴)⟩
54sneqi 4602 . 2 {⟨0, ( I ‘( I ‘𝐴))⟩} = {⟨0, ( I ‘𝐴)⟩}
6 df-s1 14491 . 2 ⟨“( I ‘𝐴)”⟩ = {⟨0, ( I ‘( I ‘𝐴))⟩}
7 df-s1 14491 . 2 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
85, 6, 73eqtr4ri 2776 1 ⟨“𝐴”⟩ = ⟨“( I ‘𝐴)”⟩
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  Vcvv 3448  {csn 4591  cop 4597   I cid 5535  cfv 6501  0cc0 11058  ⟨“cs1 14490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-s1 14491
This theorem is referenced by:  s1prc  14499  s1cli  14500  revs1  14660
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