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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 6685 | . . . . 5 ⊢ ( I ‘𝐴) ∈ V | |
2 | fvi 6742 | . . . . 5 ⊢ (( I ‘𝐴) ∈ V → ( I ‘( I ‘𝐴)) = ( I ‘𝐴)) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ ( I ‘( I ‘𝐴)) = ( I ‘𝐴) |
4 | 3 | opeq2i 4809 | . . 3 ⊢ 〈0, ( I ‘( I ‘𝐴))〉 = 〈0, ( I ‘𝐴)〉 |
5 | 4 | sneqi 4580 | . 2 ⊢ {〈0, ( I ‘( I ‘𝐴))〉} = {〈0, ( I ‘𝐴)〉} |
6 | df-s1 13952 | . 2 ⊢ 〈“( I ‘𝐴)”〉 = {〈0, ( I ‘( I ‘𝐴))〉} | |
7 | df-s1 13952 | . 2 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
8 | 5, 6, 7 | 3eqtr4ri 2857 | 1 ⊢ 〈“𝐴”〉 = 〈“( I ‘𝐴)”〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3496 {csn 4569 〈cop 4575 I cid 5461 ‘cfv 6357 0cc0 10539 〈“cs1 13951 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-s1 13952 |
This theorem is referenced by: s1prc 13960 s1cli 13961 revs1 14129 |
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