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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 6904 | . . . . 5 ⊢ ( I ‘𝐴) ∈ V | |
2 | fvi 6967 | . . . . 5 ⊢ (( I ‘𝐴) ∈ V → ( I ‘( I ‘𝐴)) = ( I ‘𝐴)) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ ( I ‘( I ‘𝐴)) = ( I ‘𝐴) |
4 | 3 | opeq2i 4877 | . . 3 ⊢ 〈0, ( I ‘( I ‘𝐴))〉 = 〈0, ( I ‘𝐴)〉 |
5 | 4 | sneqi 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)〉} | |
8 | 5, 6, 7 | 3eqtr4ri 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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