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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 36035 | . . . 4 ⊢ (𝑥 = 𝑅 → (dom 𝑥 / 𝑥) = (dom 𝑅 / 𝑅)) | |
2 | id 22 | . . . 4 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴) | |
3 | 1, 2 | eqeqan12d 2815 | . . 3 ⊢ ((𝑥 = 𝑅 ∧ 𝑦 = 𝐴) → ((dom 𝑥 / 𝑥) = 𝑦 ↔ (dom 𝑅 / 𝑅) = 𝐴)) |
4 | df-dmqss 36033 | . . 3 ⊢ DomainQss = {〈𝑥, 𝑦〉 ∣ (dom 𝑥 / 𝑥) = 𝑦} | |
5 | 3, 4 | brabga 5386 | . 2 ⊢ ((𝑅 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)) |
6 | 5 | ancoms 462 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 dom cdm 5519 / cqs 8271 DomainQss cdmqss 35636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ec 8274 df-qs 8278 df-dmqss 36033 |
This theorem is referenced by: brdmqssqs 36042 cnvepresdmqss 36046 |
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