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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 38582 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦 ↔ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦))) | |
| 2 | 1 | el2v 3443 | . . . 4 ⊢ (𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦 ↔ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)) |
| 3 | 2 | opabbii 5160 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦} = {〈𝑥, 𝑦〉 ∣ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)} |
| 4 | inopab 5774 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦}) = {〈𝑥, 𝑦〉 ∣ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)} | |
| 5 | 3, 4 | eqtr4i 2757 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦} = ({〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦}) |
| 6 | relcoss 38531 | . . 3 ⊢ Rel ≀ ◡(𝐴 ⋉ 𝐵) | |
| 7 | dfrel4v 6143 | . . 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 38531 | . . . 4 ⊢ Rel ≀ ◡𝐴 | |
| 10 | dfrel4v 6143 | . . . 4 ⊢ (Rel ≀ ◡𝐴 ↔ ≀ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦}) | |
| 11 | 9, 10 | mpbi 230 | . . 3 ⊢ ≀ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦} |
| 12 | relcoss 38531 | . . . 4 ⊢ Rel ≀ ◡𝐵 | |
| 13 | dfrel4v 6143 | . . . 4 ⊢ (Rel ≀ ◡𝐵 ↔ ≀ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦}) | |
| 14 | 12, 13 | mpbi 230 | . . 3 ⊢ ≀ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦} |
| 15 | 11, 14 | ineq12i 4167 | . 2 ⊢ ( ≀ ◡𝐴 ∩ ≀ ◡𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦}) |
| 16 | 5, 8, 15 | 3eqtr4i 2764 | 1 ⊢ ≀ ◡(𝐴 ⋉ 𝐵) = ( ≀ ◡𝐴 ∩ ≀ ◡𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1541 Vcvv 3436 ∩ cin 3896 class class class wbr 5093 {copab 5155 ◡ccnv 5618 Rel wrel 5624 ⋉ cxrn 38220 ≀ ccoss 38228 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5236 ax-nul 5246 ax-pr 5372 ax-un 7674 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4283 df-if 4475 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-fo 6493 df-fv 6495 df-1st 7927 df-2nd 7928 df-ec 8630 df-xrn 38410 df-coss 38519 |
| This theorem is referenced by: disjxrn 38850 |
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