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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 6906 | . . . 4 ⊢ (𝐴 = 𝐵 → ( I ‘𝐴) = ( I ‘𝐵)) | |
| 2 | 1 | opeq2d 4880 | . . 3 ⊢ (𝐴 = 𝐵 → 〈0, ( I ‘𝐴)〉 = 〈0, ( I ‘𝐵)〉) |
| 3 | 2 | sneqd 4638 | . 2 ⊢ (𝐴 = 𝐵 → {〈0, ( I ‘𝐴)〉} = {〈0, ( I ‘𝐵)〉}) |
| 4 | df-s1 14634 | . 2 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
| 5 | df-s1 14634 | . 2 ⊢ 〈“𝐵”〉 = {〈0, ( I ‘𝐵)〉} | |
| 6 | 3, 4, 5 | 3eqtr4g 2802 | 1 ⊢ (𝐴 = 𝐵 → 〈“𝐴”〉 = 〈“𝐵”〉) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 {csn 4626 〈cop 4632 I cid 5577 ‘cfv 6561 0cc0 11155 〈“cs1 14633 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-s1 14634 |
| This theorem is referenced by: s1eqd 14639 wrdl1exs1 14651 wrdl1s1 14652 ccats1pfxeqrex 14753 wrdind 14760 wrd2ind 14761 reuccatpfxs1lem 14784 reuccatpfxs1 14785 revs1 14803 vrmdval 18870 frgpup3lem 19795 vdegp1ci 29556 clwwlknonwwlknonb 30125 mrsubcv 35515 mrsubrn 35518 elmrsubrn 35525 mrsubvrs 35527 mvhval 35539 |
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