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Mirrors > Home > MPE Home > Th. List > Mathboxes > erALTVeq1 | Structured version Visualization version GIF version |
Description: Equality theorem for equivalence relation on domain quotient. (Contributed by Peter Mazsa, 25-Sep-2021.) |
Ref | Expression |
---|---|
erALTVeq1 | ⊢ (𝑅 = 𝑆 → (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvreleq 38074 | . . 3 ⊢ (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆)) | |
2 | dmqseqeq1 38115 | . . 3 ⊢ (𝑅 = 𝑆 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴)) | |
3 | 1, 2 | anbi12d 631 | . 2 ⊢ (𝑅 = 𝑆 → (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( EqvRel 𝑆 ∧ (dom 𝑆 / 𝑆) = 𝐴))) |
4 | dferALTV2 38140 | . 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) | |
5 | dferALTV2 38140 | . 2 ⊢ (𝑆 ErALTV 𝐴 ↔ ( EqvRel 𝑆 ∧ (dom 𝑆 / 𝑆) = 𝐴)) | |
6 | 3, 4, 5 | 3bitr4g 314 | 1 ⊢ (𝑅 = 𝑆 → (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 dom cdm 5678 / cqs 8723 EqvRel weqvrel 37665 ErALTV werALTV 37674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ec 8726 df-qs 8730 df-refrel 37984 df-symrel 38016 df-trrel 38046 df-eqvrel 38057 df-dmqs 38111 df-erALTV 38136 |
This theorem is referenced by: erALTVeq1i 38142 erALTVeq1d 38143 |
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