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Mirrors > Home > MPE Home > Th. List > Mathboxes > br1cossxrncnvssrres | Structured version Visualization version GIF version |
Description: 〈𝐵, 𝐶〉 and 〈𝐷, 𝐸〉 are cosets by range Cartesian product with restricted converse subsets class: a binary relation. (Contributed by Peter Mazsa, 9-Jun-2021.) |
Ref | Expression |
---|---|
br1cossxrncnvssrres | ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (〈𝐵, 𝐶〉 ≀ (𝑅 ⋉ (◡ S ↾ 𝐴))〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 ((𝐶 ⊆ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ⊆ 𝑢 ∧ 𝑢𝑅𝐷)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | br1cossxrnres 35568 | . 2 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (〈𝐵, 𝐶〉 ≀ (𝑅 ⋉ (◡ S ↾ 𝐴))〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 ((𝑢◡ S 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ S 𝐸 ∧ 𝑢𝑅𝐷)))) | |
2 | brcnvssr 35626 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ S 𝐶 ↔ 𝐶 ⊆ 𝑢)) | |
3 | 2 | elv 3497 | . . . . 5 ⊢ (𝑢◡ S 𝐶 ↔ 𝐶 ⊆ 𝑢) |
4 | 3 | anbi1i 623 | . . . 4 ⊢ ((𝑢◡ S 𝐶 ∧ 𝑢𝑅𝐵) ↔ (𝐶 ⊆ 𝑢 ∧ 𝑢𝑅𝐵)) |
5 | brcnvssr 35626 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ S 𝐸 ↔ 𝐸 ⊆ 𝑢)) | |
6 | 5 | elv 3497 | . . . . 5 ⊢ (𝑢◡ S 𝐸 ↔ 𝐸 ⊆ 𝑢) |
7 | 6 | anbi1i 623 | . . . 4 ⊢ ((𝑢◡ S 𝐸 ∧ 𝑢𝑅𝐷) ↔ (𝐸 ⊆ 𝑢 ∧ 𝑢𝑅𝐷)) |
8 | 4, 7 | anbi12i 626 | . . 3 ⊢ (((𝑢◡ S 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ S 𝐸 ∧ 𝑢𝑅𝐷)) ↔ ((𝐶 ⊆ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ⊆ 𝑢 ∧ 𝑢𝑅𝐷))) |
9 | 8 | rexbii 3244 | . 2 ⊢ (∃𝑢 ∈ 𝐴 ((𝑢◡ S 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ S 𝐸 ∧ 𝑢𝑅𝐷)) ↔ ∃𝑢 ∈ 𝐴 ((𝐶 ⊆ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ⊆ 𝑢 ∧ 𝑢𝑅𝐷))) |
10 | 1, 9 | syl6bb 288 | 1 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (〈𝐵, 𝐶〉 ≀ (𝑅 ⋉ (◡ S ↾ 𝐴))〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 ((𝐶 ⊆ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ⊆ 𝑢 ∧ 𝑢𝑅𝐷)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 ∃wrex 3136 Vcvv 3492 ⊆ wss 3933 〈cop 4563 class class class wbr 5057 ◡ccnv 5547 ↾ cres 5550 ⋉ cxrn 35333 ≀ ccoss 35334 S cssr 35337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-1st 7678 df-2nd 7679 df-xrn 35503 df-coss 35539 df-ssr 35618 |
This theorem is referenced by: (None) |
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