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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 6847 | . . . 4 ⊢ (𝐴 = 𝐵 → ( I ‘𝐴) = ( I ‘𝐵)) | |
2 | 1 | opeq2d 4842 | . . 3 ⊢ (𝐴 = 𝐵 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, ( I ‘𝐵)⟩) |
3 | 2 | sneqd 4603 | . 2 ⊢ (𝐴 = 𝐵 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, ( I ‘𝐵)⟩}) |
4 | df-s1 14491 | . 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩} | |
5 | df-s1 14491 | . 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 1542 {csn 4591 ⟨cop 4597 I cid 5535 ‘cfv 6501 0cc0 11058 ⟨“cs1 14490 |
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 2708 |
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 2715 df-cleq 2729 df-clel 2815 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-iota 6453 df-fv 6509 df-s1 14491 |
This theorem is referenced by: s1eqd 14496 wrdl1exs1 14508 wrdl1s1 14509 ccats1pfxeqrex 14610 wrdind 14617 wrd2ind 14618 reuccatpfxs1lem 14641 reuccatpfxs1 14642 revs1 14660 vrmdval 18674 frgpup3lem 19566 vdegp1ci 28528 clwwlknonwwlknonb 29092 mrsubcv 34144 mrsubrn 34147 elmrsubrn 34154 mrsubvrs 34156 mvhval 34168 |
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