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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 6687 | . . . . 5 ⊢ ( I ‘𝐴) ∈ V | |
2 | fvi 6744 | . . . . 5 ⊢ (( I ‘𝐴) ∈ V → ( I ‘( I ‘𝐴)) = ( I ‘𝐴)) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ ( I ‘( I ‘𝐴)) = ( I ‘𝐴) |
4 | 3 | opeq2i 4765 | . . 3 ⊢ 〈0, ( I ‘( I ‘𝐴))〉 = 〈0, ( I ‘𝐴)〉 |
5 | 4 | sneqi 4527 | . 2 ⊢ {〈0, ( I ‘( I ‘𝐴))〉} = {〈0, ( I ‘𝐴)〉} |
6 | df-s1 14039 | . 2 ⊢ 〈“( I ‘𝐴)”〉 = {〈0, ( I ‘( I ‘𝐴))〉} | |
7 | df-s1 14039 | . 2 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
8 | 5, 6, 7 | 3eqtr4ri 2772 | 1 ⊢ 〈“𝐴”〉 = 〈“( I ‘𝐴)”〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2114 Vcvv 3398 {csn 4516 〈cop 4522 I cid 5428 ‘cfv 6339 0cc0 10615 〈“cs1 14038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-iota 6297 df-fun 6341 df-fv 6347 df-s1 14039 |
This theorem is referenced by: s1prc 14047 s1cli 14048 revs1 14216 |
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