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Mirrors > Home > MPE Home > Th. List > Mathboxes > releldmqs | Structured version Visualization version GIF version |
Description: Elementhood in the domain quotient of a relation. (Contributed by Peter Mazsa, 24-Apr-2021.) |
Ref | Expression |
---|---|
releldmqs | ⊢ (𝐴 ∈ 𝑉 → (Rel 𝑅 → (𝐴 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑢]𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resdm 6046 | . . . . . 6 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅) | |
2 | 1 | dmqseqd 38624 | . . . . 5 ⊢ (Rel 𝑅 → (dom (𝑅 ↾ dom 𝑅) / (𝑅 ↾ dom 𝑅)) = (dom 𝑅 / 𝑅)) |
3 | 2 | eleq2d 2825 | . . . 4 ⊢ (Rel 𝑅 → (𝐴 ∈ (dom (𝑅 ↾ dom 𝑅) / (𝑅 ↾ dom 𝑅)) ↔ 𝐴 ∈ (dom 𝑅 / 𝑅))) |
4 | 3 | adantl 481 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Rel 𝑅) → (𝐴 ∈ (dom (𝑅 ↾ dom 𝑅) / (𝑅 ↾ dom 𝑅)) ↔ 𝐴 ∈ (dom 𝑅 / 𝑅))) |
5 | eldmqsres2 38270 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (dom (𝑅 ↾ dom 𝑅) / (𝑅 ↾ dom 𝑅)) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑢]𝑅)) | |
6 | 5 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Rel 𝑅) → (𝐴 ∈ (dom (𝑅 ↾ dom 𝑅) / (𝑅 ↾ dom 𝑅)) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑢]𝑅)) |
7 | 4, 6 | bitr3d 281 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Rel 𝑅) → (𝐴 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑢]𝑅)) |
8 | 7 | ex 412 | 1 ⊢ (𝐴 ∈ 𝑉 → (Rel 𝑅 → (𝐴 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑢]𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 dom cdm 5689 ↾ cres 5691 Rel wrel 5694 [cec 8742 / cqs 8743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ec 8746 df-qs 8750 |
This theorem is referenced by: disjdmqsss 38784 |
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