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Theorem ids1 14574
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 6905 . . . . 5 ( I ‘𝐴) ∈ V
2 fvi 6969 . . . . 5 (( I ‘𝐴) ∈ V → ( I ‘( I ‘𝐴)) = ( I ‘𝐴))
31, 2ax-mp 5 . . . 4 ( I ‘( I ‘𝐴)) = ( I ‘𝐴)
43opeq2i 4874 . . 3 ⟨0, ( I ‘( I ‘𝐴))⟩ = ⟨0, ( I ‘𝐴)⟩
54sneqi 4636 . 2 {⟨0, ( I ‘( I ‘𝐴))⟩} = {⟨0, ( I ‘𝐴)⟩}
6 df-s1 14573 . 2 ⟨“( I ‘𝐴)”⟩ = {⟨0, ( I ‘( I ‘𝐴))⟩}
7 df-s1 14573 . 2 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
85, 6, 73eqtr4ri 2767 1 ⟨“𝐴”⟩ = ⟨“( I ‘𝐴)”⟩
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wcel 2099  Vcvv 3470  {csn 4625  cop 4631   I cid 5570  cfv 6543  0cc0 11133  ⟨“cs1 14572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-s1 14573
This theorem is referenced by:  s1prc  14581  s1cli  14582  revs1  14742
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