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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 36956 | . 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 37014 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ S 𝐶 ↔ 𝐶 ⊆ 𝑢)) | |
3 | 2 | elv 3450 | . . . . 5 ⊢ (𝑢◡ S 𝐶 ↔ 𝐶 ⊆ 𝑢) |
4 | 3 | anbi1i 625 | . . . 4 ⊢ ((𝑢◡ S 𝐶 ∧ 𝑢𝑅𝐵) ↔ (𝐶 ⊆ 𝑢 ∧ 𝑢𝑅𝐵)) |
5 | brcnvssr 37014 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ S 𝐸 ↔ 𝐸 ⊆ 𝑢)) | |
6 | 5 | elv 3450 | . . . . 5 ⊢ (𝑢◡ S 𝐸 ↔ 𝐸 ⊆ 𝑢) |
7 | 6 | anbi1i 625 | . . . 4 ⊢ ((𝑢◡ S 𝐸 ∧ 𝑢𝑅𝐷) ↔ (𝐸 ⊆ 𝑢 ∧ 𝑢𝑅𝐷)) |
8 | 4, 7 | anbi12i 628 | . . 3 ⊢ (((𝑢◡ S 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ S 𝐸 ∧ 𝑢𝑅𝐷)) ↔ ((𝐶 ⊆ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ⊆ 𝑢 ∧ 𝑢𝑅𝐷))) |
9 | 8 | rexbii 3094 | . 2 ⊢ (∃𝑢 ∈ 𝐴 ((𝑢◡ S 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ S 𝐸 ∧ 𝑢𝑅𝐷)) ↔ ∃𝑢 ∈ 𝐴 ((𝐶 ⊆ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ⊆ 𝑢 ∧ 𝑢𝑅𝐷))) |
10 | 1, 9 | bitrdi 287 | 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 205 ∧ wa 397 ∈ wcel 2107 ∃wrex 3070 Vcvv 3444 ⊆ wss 3911 ⟨cop 4593 class class class wbr 5106 ◡ccnv 5633 ↾ cres 5636 ⋉ cxrn 36679 ≀ ccoss 36680 S cssr 36683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fo 6503 df-fv 6505 df-1st 7922 df-2nd 7923 df-xrn 36879 df-coss 36919 df-ssr 37006 |
This theorem is referenced by: (None) |
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