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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 6352 | . . . 4 ⊢ (𝐴 = 𝐵 → ( I ‘𝐴) = ( I ‘𝐵)) | |
2 | 1 | opeq2d 4560 | . . 3 ⊢ (𝐴 = 𝐵 → 〈0, ( I ‘𝐴)〉 = 〈0, ( I ‘𝐵)〉) |
3 | 2 | sneqd 4333 | . 2 ⊢ (𝐴 = 𝐵 → {〈0, ( I ‘𝐴)〉} = {〈0, ( I ‘𝐵)〉}) |
4 | df-s1 13488 | . 2 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
5 | df-s1 13488 | . 2 ⊢ 〈“𝐵”〉 = {〈0, ( I ‘𝐵)〉} | |
6 | 3, 4, 5 | 3eqtr4g 2819 | 1 ⊢ (𝐴 = 𝐵 → 〈“𝐴”〉 = 〈“𝐵”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 {csn 4321 〈cop 4327 I cid 5173 ‘cfv 6049 0cc0 10128 〈“cs1 13480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-rex 3056 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-iota 6012 df-fv 6057 df-s1 13488 |
This theorem is referenced by: s1eqd 13571 wrdl1exs1 13584 wrdl1s1 13585 wrdind 13676 wrd2ind 13677 ccats1swrdeqrex 13678 reuccats1lem 13679 reuccats1 13680 revs1 13714 vrmdval 17595 frgpup3lem 18390 vdegp1ci 26644 clwwlknonwwlknonb 27254 clwwlknonwwlknonbOLD 27255 mrsubcv 31714 mrsubrn 31717 elmrsubrn 31724 mrsubvrs 31726 mvhval 31738 ccats1pfxeqrex 41932 reuccatpfxs1lem 41943 reuccatpfxs1 41944 |
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