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Mirrors > Home > MPE Home > Th. List > ids1 | Structured version Visualization version GIF version |
Description: Identity function protection for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
Ref | Expression |
---|---|
ids1 | ⊢ ⟨“𝐴”⟩ = ⟨“( I ‘𝐴)”⟩ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6905 | . . . . 5 ⊢ ( I ‘𝐴) ∈ V | |
2 | fvi 6969 | . . . . 5 ⊢ (( I ‘𝐴) ∈ V → ( I ‘( I ‘𝐴)) = ( I ‘𝐴)) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ ( I ‘( I ‘𝐴)) = ( I ‘𝐴) |
4 | 3 | opeq2i 4874 | . . 3 ⊢ ⟨0, ( I ‘( I ‘𝐴))⟩ = ⟨0, ( I ‘𝐴)⟩ |
5 | 4 | sneqi 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)⟩} | |
8 | 5, 6, 7 | 3eqtr4ri 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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