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Mirrors > Home > MPE Home > Th. List > Mathboxes > brcoels | Structured version Visualization version GIF version |
Description: 𝐵 and 𝐶 are coelements : a binary relation. (Contributed by Peter Mazsa, 14-Jan-2020.) (Revised by Peter Mazsa, 5-Oct-2021.) |
Ref | Expression |
---|---|
brcoels | ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ∼ 𝐴𝐶 ↔ ∃𝑢 ∈ 𝐴 (𝐵 ∈ 𝑢 ∧ 𝐶 ∈ 𝑢))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2827 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝑢 ↔ 𝐵 ∈ 𝑢)) | |
2 | eleq1 2827 | . . . 4 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ 𝑢 ↔ 𝐶 ∈ 𝑢)) | |
3 | 1, 2 | bi2anan9 638 | . . 3 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ (𝐵 ∈ 𝑢 ∧ 𝐶 ∈ 𝑢))) |
4 | 3 | rexbidv 3177 | . 2 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → (∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) ↔ ∃𝑢 ∈ 𝐴 (𝐵 ∈ 𝑢 ∧ 𝐶 ∈ 𝑢))) |
5 | dfcoels 38412 | . 2 ⊢ ∼ 𝐴 = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} | |
6 | 4, 5 | brabga 5544 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ∼ 𝐴𝐶 ↔ ∃𝑢 ∈ 𝐴 (𝐵 ∈ 𝑢 ∧ 𝐶 ∈ 𝑢))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 class class class wbr 5148 ∼ ccoels 38163 |
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-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-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 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-eprel 5589 df-xp 5695 df-rel 5696 df-cnv 5697 df-res 5701 df-coss 38393 df-coels 38394 |
This theorem is referenced by: erimeq2 38660 |
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