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Mirrors > Home > MPE Home > Th. List > Mathboxes > eldmqsres | Structured version Visualization version GIF version |
Description: Elementhood in a restricted domain quotient set. (Contributed by Peter Mazsa, 21-Aug-2020.) |
Ref | Expression |
---|---|
eldmqsres | ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom (𝑅 ↾ 𝐴) / (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elqsg 8785 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom (𝑅 ↾ 𝐴) / (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ dom (𝑅 ↾ 𝐴)𝐵 = [𝑢](𝑅 ↾ 𝐴))) | |
2 | eldmres2 37803 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 ∈ dom (𝑅 ↾ 𝐴) ↔ (𝑢 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅))) | |
3 | 2 | elv 3469 | . . . . 5 ⊢ (𝑢 ∈ dom (𝑅 ↾ 𝐴) ↔ (𝑢 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅)) |
4 | 3 | anbi1i 622 | . . . 4 ⊢ ((𝑢 ∈ dom (𝑅 ↾ 𝐴) ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴)) ↔ ((𝑢 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴))) |
5 | ecres2 37807 | . . . . . . . 8 ⊢ (𝑢 ∈ 𝐴 → [𝑢](𝑅 ↾ 𝐴) = [𝑢]𝑅) | |
6 | 5 | eqeq2d 2736 | . . . . . . 7 ⊢ (𝑢 ∈ 𝐴 → (𝐵 = [𝑢](𝑅 ↾ 𝐴) ↔ 𝐵 = [𝑢]𝑅)) |
7 | 6 | pm5.32i 573 | . . . . . 6 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴)) ↔ (𝑢 ∈ 𝐴 ∧ 𝐵 = [𝑢]𝑅)) |
8 | 7 | anbi2i 621 | . . . . 5 ⊢ ((∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢 ∈ 𝐴 ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴))) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢 ∈ 𝐴 ∧ 𝐵 = [𝑢]𝑅))) |
9 | an21 642 | . . . . 5 ⊢ (((𝑢 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴)) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢 ∈ 𝐴 ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴)))) | |
10 | an12 643 | . . . . 5 ⊢ ((𝑢 ∈ 𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅)) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢 ∈ 𝐴 ∧ 𝐵 = [𝑢]𝑅))) | |
11 | 8, 9, 10 | 3bitr4i 302 | . . . 4 ⊢ (((𝑢 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴)) ↔ (𝑢 ∈ 𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅))) |
12 | 4, 11 | bitri 274 | . . 3 ⊢ ((𝑢 ∈ dom (𝑅 ↾ 𝐴) ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴)) ↔ (𝑢 ∈ 𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅))) |
13 | 12 | rexbii2 3080 | . 2 ⊢ (∃𝑢 ∈ dom (𝑅 ↾ 𝐴)𝐵 = [𝑢](𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅)) |
14 | 1, 13 | bitrdi 286 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom (𝑅 ↾ 𝐴) / (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∃wrex 3060 Vcvv 3463 dom cdm 5672 ↾ cres 5674 [cec 8721 / cqs 8722 |
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-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5144 df-opab 5206 df-xp 5678 df-rel 5679 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ec 8725 df-qs 8729 |
This theorem is referenced by: eldmqsres2 37816 |
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