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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 6695 | . . . 4 ⊢ (𝐴 = 𝐵 → ( I ‘𝐴) = ( I ‘𝐵)) | |
2 | 1 | opeq2d 4777 | . . 3 ⊢ (𝐴 = 𝐵 → 〈0, ( I ‘𝐴)〉 = 〈0, ( I ‘𝐵)〉) |
3 | 2 | sneqd 4539 | . 2 ⊢ (𝐴 = 𝐵 → {〈0, ( I ‘𝐴)〉} = {〈0, ( I ‘𝐵)〉}) |
4 | df-s1 14118 | . 2 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
5 | df-s1 14118 | . 2 ⊢ 〈“𝐵”〉 = {〈0, ( I ‘𝐵)〉} | |
6 | 3, 4, 5 | 3eqtr4g 2796 | 1 ⊢ (𝐴 = 𝐵 → 〈“𝐴”〉 = 〈“𝐵”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 {csn 4527 〈cop 4533 I cid 5439 ‘cfv 6358 0cc0 10694 〈“cs1 14117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-iota 6316 df-fv 6366 df-s1 14118 |
This theorem is referenced by: s1eqd 14123 wrdl1exs1 14135 wrdl1s1 14136 ccats1pfxeqrex 14245 wrdind 14252 wrd2ind 14253 reuccatpfxs1lem 14276 reuccatpfxs1 14277 revs1 14295 vrmdval 18238 frgpup3lem 19121 vdegp1ci 27580 clwwlknonwwlknonb 28143 mrsubcv 33139 mrsubrn 33142 elmrsubrn 33149 mrsubvrs 33151 mvhval 33163 |
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