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Theorem ids1 13687
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 6459 . . . . 5 ( I ‘𝐴) ∈ V
2 fvi 6515 . . . . 5 (( I ‘𝐴) ∈ V → ( I ‘( I ‘𝐴)) = ( I ‘𝐴))
31, 2ax-mp 5 . . . 4 ( I ‘( I ‘𝐴)) = ( I ‘𝐴)
43opeq2i 4640 . . 3 ⟨0, ( I ‘( I ‘𝐴))⟩ = ⟨0, ( I ‘𝐴)⟩
54sneqi 4409 . 2 {⟨0, ( I ‘( I ‘𝐴))⟩} = {⟨0, ( I ‘𝐴)⟩}
6 df-s1 13686 . 2 ⟨“( I ‘𝐴)”⟩ = {⟨0, ( I ‘( I ‘𝐴))⟩}
7 df-s1 13686 . 2 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
85, 6, 73eqtr4ri 2813 1 ⟨“𝐴”⟩ = ⟨“( I ‘𝐴)”⟩
Colors of variables: wff setvar class
Syntax hints:   = wceq 1601  wcel 2107  Vcvv 3398  {csn 4398  cop 4404   I cid 5260  cfv 6135  0cc0 10272  ⟨“cs1 13685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-s1 13686
This theorem is referenced by:  s1prc  13694  s1cli  13695  revs1  13911
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