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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 38022 | . . . 4 ⊢ (𝑥 = 𝑅 → (dom 𝑥 / 𝑥) = (dom 𝑅 / 𝑅)) | |
2 | id 22 | . . . 4 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴) | |
3 | 1, 2 | eqeqan12d 2740 | . . 3 ⊢ ((𝑥 = 𝑅 ∧ 𝑦 = 𝐴) → ((dom 𝑥 / 𝑥) = 𝑦 ↔ (dom 𝑅 / 𝑅) = 𝐴)) |
4 | df-dmqss 38020 | . . 3 ⊢ DomainQss = {⟨𝑥, 𝑦⟩ ∣ (dom 𝑥 / 𝑥) = 𝑦} | |
5 | 3, 4 | brabga 5527 | . 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 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 class class class wbr 5141 dom cdm 5669 / cqs 8701 DomainQss cdmqss 37578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ec 8704 df-qs 8708 df-dmqss 38020 |
This theorem is referenced by: brdmqssqs 38029 cnvepresdmqss 38034 brparts2 38154 |
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