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Mirrors > Home > MPE Home > Th. List > Mathboxes > br1cossxrnidres | Structured version Visualization version GIF version |
Description: 〈𝐵, 𝐶〉 and 〈𝐷, 𝐸〉 are cosets by a range Cartesian product with the restricted identity class: a binary relation. (Contributed by Peter Mazsa, 8-Jun-2021.) |
Ref | Expression |
---|---|
br1cossxrnidres | ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (〈𝐵, 𝐶〉 ≀ (𝑅 ⋉ ( I ↾ 𝐴))〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 ((𝑢 = 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐸 ∧ 𝑢𝑅𝐷)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | br1cossxrnres 38429 | . 2 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (〈𝐵, 𝐶〉 ≀ (𝑅 ⋉ ( I ↾ 𝐴))〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 ((𝑢 I 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 I 𝐸 ∧ 𝑢𝑅𝐷)))) | |
2 | ideq2 38288 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 I 𝐶 ↔ 𝑢 = 𝐶)) | |
3 | 2 | elv 3482 | . . . . 5 ⊢ (𝑢 I 𝐶 ↔ 𝑢 = 𝐶) |
4 | 3 | anbi1i 624 | . . . 4 ⊢ ((𝑢 I 𝐶 ∧ 𝑢𝑅𝐵) ↔ (𝑢 = 𝐶 ∧ 𝑢𝑅𝐵)) |
5 | ideq2 38288 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 I 𝐸 ↔ 𝑢 = 𝐸)) | |
6 | 5 | elv 3482 | . . . . 5 ⊢ (𝑢 I 𝐸 ↔ 𝑢 = 𝐸) |
7 | 6 | anbi1i 624 | . . . 4 ⊢ ((𝑢 I 𝐸 ∧ 𝑢𝑅𝐷) ↔ (𝑢 = 𝐸 ∧ 𝑢𝑅𝐷)) |
8 | 4, 7 | anbi12i 628 | . . 3 ⊢ (((𝑢 I 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 I 𝐸 ∧ 𝑢𝑅𝐷)) ↔ ((𝑢 = 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐸 ∧ 𝑢𝑅𝐷))) |
9 | 8 | rexbii 3091 | . 2 ⊢ (∃𝑢 ∈ 𝐴 ((𝑢 I 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 I 𝐸 ∧ 𝑢𝑅𝐷)) ↔ ∃𝑢 ∈ 𝐴 ((𝑢 = 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐸 ∧ 𝑢𝑅𝐷))) |
10 | 1, 9 | bitrdi 287 | 1 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (〈𝐵, 𝐶〉 ≀ (𝑅 ⋉ ( I ↾ 𝐴))〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 ((𝑢 = 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐸 ∧ 𝑢𝑅𝐷)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 Vcvv 3477 〈cop 4636 class class class wbr 5147 I cid 5581 ↾ cres 5690 ⋉ cxrn 38160 ≀ ccoss 38161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fo 6568 df-fv 6570 df-1st 8012 df-2nd 8013 df-xrn 38352 df-coss 38392 |
This theorem is referenced by: (None) |
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