![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4884 | . . 3 ⊢ (𝐴 = 𝐵 → 〈0, ( I ‘𝐴)〉 = 〈0, ( I ‘𝐵)〉) |
3 | 2 | sneqd 4642 | . 2 ⊢ (𝐴 = 𝐵 → {〈0, ( I ‘𝐴)〉} = {〈0, ( I ‘𝐵)〉}) |
4 | df-s1 14630 | . 2 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
5 | df-s1 14630 | . 2 ⊢ 〈“𝐵”〉 = {〈0, ( I ‘𝐵)〉} | |
6 | 3, 4, 5 | 3eqtr4g 2799 | 1 ⊢ (𝐴 = 𝐵 → 〈“𝐴”〉 = 〈“𝐵”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 {csn 4630 〈cop 4636 I cid 5581 ‘cfv 6562 0cc0 11152 〈“cs1 14629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-iota 6515 df-fv 6570 df-s1 14630 |
This theorem is referenced by: s1eqd 14635 wrdl1exs1 14647 wrdl1s1 14648 ccats1pfxeqrex 14749 wrdind 14756 wrd2ind 14757 reuccatpfxs1lem 14780 reuccatpfxs1 14781 revs1 14799 vrmdval 18882 frgpup3lem 19809 vdegp1ci 29570 clwwlknonwwlknonb 30134 mrsubcv 35494 mrsubrn 35497 elmrsubrn 35504 mrsubvrs 35506 mvhval 35518 |
Copyright terms: Public domain | W3C validator |