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Mirrors > Home > MPE Home > Th. List > s1eq | Structured version Visualization version GIF version |
Description: Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
Ref | Expression |
---|---|
s1eq | ⊢ (𝐴 = 𝐵 → 〈“𝐴”〉 = 〈“𝐵”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6672 | . . . 4 ⊢ (𝐴 = 𝐵 → ( I ‘𝐴) = ( I ‘𝐵)) | |
2 | 1 | opeq2d 4812 | . . 3 ⊢ (𝐴 = 𝐵 → 〈0, ( I ‘𝐴)〉 = 〈0, ( I ‘𝐵)〉) |
3 | 2 | sneqd 4581 | . 2 ⊢ (𝐴 = 𝐵 → {〈0, ( I ‘𝐴)〉} = {〈0, ( I ‘𝐵)〉}) |
4 | df-s1 13952 | . 2 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
5 | df-s1 13952 | . 2 ⊢ 〈“𝐵”〉 = {〈0, ( I ‘𝐵)〉} | |
6 | 3, 4, 5 | 3eqtr4g 2883 | 1 ⊢ (𝐴 = 𝐵 → 〈“𝐴”〉 = 〈“𝐵”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 {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 |
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-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 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-iota 6316 df-fv 6365 df-s1 13952 |
This theorem is referenced by: s1eqd 13957 wrdl1exs1 13969 wrdl1s1 13970 ccats1pfxeqrex 14079 wrdind 14086 wrd2ind 14087 reuccatpfxs1lem 14110 reuccatpfxs1 14111 revs1 14129 vrmdval 18024 frgpup3lem 18905 vdegp1ci 27322 clwwlknonwwlknonb 27887 mrsubcv 32759 mrsubrn 32762 elmrsubrn 32769 mrsubvrs 32771 mvhval 32783 |
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