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Theorem ids1 13089
 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 5997 . . . . 5 ( I ‘𝐴) ∈ V
2 fvi 6049 . . . . 5 (( I ‘𝐴) ∈ V → ( I ‘( I ‘𝐴)) = ( I ‘𝐴))
31, 2ax-mp 5 . . . 4 ( I ‘( I ‘𝐴)) = ( I ‘𝐴)
43opeq2i 4242 . . 3 ⟨0, ( I ‘( I ‘𝐴))⟩ = ⟨0, ( I ‘𝐴)⟩
54sneqi 4039 . 2 {⟨0, ( I ‘( I ‘𝐴))⟩} = {⟨0, ( I ‘𝐴)⟩}
6 df-s1 13016 . 2 ⟨“( I ‘𝐴)”⟩ = {⟨0, ( I ‘( I ‘𝐴))⟩}
7 df-s1 13016 . 2 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
85, 6, 73eqtr4ri 2547 1 ⟨“𝐴”⟩ = ⟨“( I ‘𝐴)”⟩
 Colors of variables: wff setvar class Syntax hints:   = wceq 1474   ∈ wcel 1938  Vcvv 3077  {csn 4028  ⟨cop 4034   I cid 4842  ‘cfv 5689  0cc0 9691  ⟨“cs1 13008 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732 This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-iota 5653  df-fun 5691  df-fv 5697  df-s1 13016 This theorem is referenced by:  s1cli  13096  revs1  13224
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