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Mirrors > Home > MPE Home > Th. List > Mathboxes > br1cossxrncnvepres | Structured version Visualization version GIF version |
Description: 〈𝐵, 𝐶〉 and 〈𝐷, 𝐸〉 are cosets by a range Cartesian product with the restricted converse epsilon class: a binary relation. (Contributed by Peter Mazsa, 12-May-2021.) |
Ref | Expression |
---|---|
br1cossxrncnvepres | ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (〈𝐵, 𝐶〉 ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 ((𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ∈ 𝑢 ∧ 𝑢𝑅𝐷)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | br1cossxrnres 36545 | . 2 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (〈𝐵, 𝐶〉 ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 ((𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐸 ∧ 𝑢𝑅𝐷)))) | |
2 | brcnvep 36383 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝐶 ↔ 𝐶 ∈ 𝑢)) | |
3 | 2 | elv 3436 | . . . . 5 ⊢ (𝑢◡ E 𝐶 ↔ 𝐶 ∈ 𝑢) |
4 | 3 | anbi1i 623 | . . . 4 ⊢ ((𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐵) ↔ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐵)) |
5 | brcnvep 36383 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝐸 ↔ 𝐸 ∈ 𝑢)) | |
6 | 5 | elv 3436 | . . . . 5 ⊢ (𝑢◡ E 𝐸 ↔ 𝐸 ∈ 𝑢) |
7 | 6 | anbi1i 623 | . . . 4 ⊢ ((𝑢◡ E 𝐸 ∧ 𝑢𝑅𝐷) ↔ (𝐸 ∈ 𝑢 ∧ 𝑢𝑅𝐷)) |
8 | 4, 7 | anbi12i 626 | . . 3 ⊢ (((𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐸 ∧ 𝑢𝑅𝐷)) ↔ ((𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ∈ 𝑢 ∧ 𝑢𝑅𝐷))) |
9 | 8 | rexbii 3179 | . 2 ⊢ (∃𝑢 ∈ 𝐴 ((𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐸 ∧ 𝑢𝑅𝐷)) ↔ ∃𝑢 ∈ 𝐴 ((𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ∈ 𝑢 ∧ 𝑢𝑅𝐷))) |
10 | 1, 9 | bitrdi 286 | 1 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (〈𝐵, 𝐶〉 ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 ((𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ∈ 𝑢 ∧ 𝑢𝑅𝐷)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2109 ∃wrex 3066 Vcvv 3430 〈cop 4572 class class class wbr 5078 E cep 5493 ◡ccnv 5587 ↾ cres 5590 ⋉ cxrn 36311 ≀ ccoss 36312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pr 5355 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-eprel 5494 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-fo 6436 df-fv 6438 df-1st 7817 df-2nd 7818 df-xrn 36480 df-coss 36516 |
This theorem is referenced by: (None) |
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