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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 6892 | . . . 4 ⊢ (𝐴 = 𝐵 → ( I ‘𝐴) = ( I ‘𝐵)) | |
2 | 1 | opeq2d 4881 | . . 3 ⊢ (𝐴 = 𝐵 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, ( I ‘𝐵)⟩) |
3 | 2 | sneqd 4641 | . 2 ⊢ (𝐴 = 𝐵 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, ( I ‘𝐵)⟩}) |
4 | df-s1 14546 | . 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩} | |
5 | df-s1 14546 | . 2 ⊢ ⟨“𝐵”⟩ = {⟨0, ( I ‘𝐵)⟩} | |
6 | 3, 4, 5 | 3eqtr4g 2798 | 1 ⊢ (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 {csn 4629 ⟨cop 4635 I cid 5574 ‘cfv 6544 0cc0 11110 ⟨“cs1 14545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 df-s1 14546 |
This theorem is referenced by: s1eqd 14551 wrdl1exs1 14563 wrdl1s1 14564 ccats1pfxeqrex 14665 wrdind 14672 wrd2ind 14673 reuccatpfxs1lem 14696 reuccatpfxs1 14697 revs1 14715 vrmdval 18738 frgpup3lem 19645 vdegp1ci 28795 clwwlknonwwlknonb 29359 mrsubcv 34501 mrsubrn 34504 elmrsubrn 34511 mrsubvrs 34513 mvhval 34525 |
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