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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 6459 | . . . . 5 ⊢ ( I ‘𝐴) ∈ V | |
2 | fvi 6515 | . . . . 5 ⊢ (( I ‘𝐴) ∈ V → ( I ‘( I ‘𝐴)) = ( I ‘𝐴)) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ ( I ‘( I ‘𝐴)) = ( I ‘𝐴) |
4 | 3 | opeq2i 4640 | . . 3 ⊢ 〈0, ( I ‘( I ‘𝐴))〉 = 〈0, ( I ‘𝐴)〉 |
5 | 4 | sneqi 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)〉} | |
8 | 5, 6, 7 | 3eqtr4ri 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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