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| Mirrors > Home > MPE Home > Th. List > Mathboxes > brdmqss | Structured version Visualization version GIF version | ||
| Description: The domain quotient binary relation. (Contributed by Peter Mazsa, 17-Apr-2019.) | 
| Ref | Expression | 
|---|---|
| brdmqss | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dmqseq 38641 | . . . 4 ⊢ (𝑥 = 𝑅 → (dom 𝑥 / 𝑥) = (dom 𝑅 / 𝑅)) | |
| 2 | id 22 | . . . 4 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴) | |
| 3 | 1, 2 | eqeqan12d 2751 | . . 3 ⊢ ((𝑥 = 𝑅 ∧ 𝑦 = 𝐴) → ((dom 𝑥 / 𝑥) = 𝑦 ↔ (dom 𝑅 / 𝑅) = 𝐴)) | 
| 4 | df-dmqss 38639 | . . 3 ⊢ DomainQss = {〈𝑥, 𝑦〉 ∣ (dom 𝑥 / 𝑥) = 𝑦} | |
| 5 | 3, 4 | brabga 5539 | . 2 ⊢ ((𝑅 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)) | 
| 6 | 5 | ancoms 458 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 dom cdm 5685 / cqs 8744 DomainQss cdmqss 38205 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ec 8747 df-qs 8751 df-dmqss 38639 | 
| This theorem is referenced by: brdmqssqs 38648 cnvepresdmqss 38653 brparts2 38773 | 
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