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Theorem ids1 14554
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 6904 . . . . 5 ( I ‘𝐴) ∈ V
2 fvi 6967 . . . . 5 (( I ‘𝐴) ∈ V → ( I ‘( I ‘𝐴)) = ( I ‘𝐴))
31, 2ax-mp 5 . . . 4 ( I ‘( I ‘𝐴)) = ( I ‘𝐴)
43opeq2i 4877 . . 3 ⟨0, ( I ‘( I ‘𝐴))⟩ = ⟨0, ( I ‘𝐴)⟩
54sneqi 4639 . 2 {⟨0, ( I ‘( I ‘𝐴))⟩} = {⟨0, ( I ‘𝐴)⟩}
6 df-s1 14553 . 2 ⟨“( I ‘𝐴)”⟩ = {⟨0, ( I ‘( I ‘𝐴))⟩}
7 df-s1 14553 . 2 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
85, 6, 73eqtr4ri 2770 1 ⟨“𝐴”⟩ = ⟨“( I ‘𝐴)”⟩
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2105  Vcvv 3473  {csn 4628  cop 4634   I cid 5573  cfv 6543  0cc0 11116  ⟨“cs1 14552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-s1 14553
This theorem is referenced by:  s1prc  14561  s1cli  14562  revs1  14722
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