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| Mirrors > Home > MPE Home > Th. List > Mathboxes > 1cosscnvxrn | Structured version Visualization version GIF version | ||
| Description: Cosets by the converse range Cartesian product. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.) | 
| Ref | Expression | 
|---|---|
| 1cosscnvxrn | ⊢ ≀ ◡(𝐴 ⋉ 𝐵) = ( ≀ ◡𝐴 ∩ ≀ ◡𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | br1cosscnvxrn 38475 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦 ↔ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦))) | |
| 2 | 1 | el2v 3487 | . . . 4 ⊢ (𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦 ↔ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)) | 
| 3 | 2 | opabbii 5210 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦} = {〈𝑥, 𝑦〉 ∣ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)} | 
| 4 | inopab 5839 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦}) = {〈𝑥, 𝑦〉 ∣ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)} | |
| 5 | 3, 4 | eqtr4i 2768 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦} = ({〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦}) | 
| 6 | relcoss 38424 | . . 3 ⊢ Rel ≀ ◡(𝐴 ⋉ 𝐵) | |
| 7 | dfrel4v 6210 | . . 3 ⊢ (Rel ≀ ◡(𝐴 ⋉ 𝐵) ↔ ≀ ◡(𝐴 ⋉ 𝐵) = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦}) | |
| 8 | 6, 7 | mpbi 230 | . 2 ⊢ ≀ ◡(𝐴 ⋉ 𝐵) = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦} | 
| 9 | relcoss 38424 | . . . 4 ⊢ Rel ≀ ◡𝐴 | |
| 10 | dfrel4v 6210 | . . . 4 ⊢ (Rel ≀ ◡𝐴 ↔ ≀ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦}) | |
| 11 | 9, 10 | mpbi 230 | . . 3 ⊢ ≀ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦} | 
| 12 | relcoss 38424 | . . . 4 ⊢ Rel ≀ ◡𝐵 | |
| 13 | dfrel4v 6210 | . . . 4 ⊢ (Rel ≀ ◡𝐵 ↔ ≀ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦}) | |
| 14 | 12, 13 | mpbi 230 | . . 3 ⊢ ≀ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦} | 
| 15 | 11, 14 | ineq12i 4218 | . 2 ⊢ ( ≀ ◡𝐴 ∩ ≀ ◡𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦}) | 
| 16 | 5, 8, 15 | 3eqtr4i 2775 | 1 ⊢ ≀ ◡(𝐴 ⋉ 𝐵) = ( ≀ ◡𝐴 ∩ ≀ ◡𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 Vcvv 3480 ∩ cin 3950 class class class wbr 5143 {copab 5205 ◡ccnv 5684 Rel wrel 5690 ⋉ cxrn 38181 ≀ ccoss 38182 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-1st 8014 df-2nd 8015 df-ec 8747 df-xrn 38372 df-coss 38412 | 
| This theorem is referenced by: disjxrn 38747 | 
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