![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
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 38456 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦 ↔ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦))) | |
2 | 1 | el2v 3485 | . . . 4 ⊢ (𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦 ↔ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)) |
3 | 2 | opabbii 5215 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦} = {〈𝑥, 𝑦〉 ∣ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)} |
4 | inopab 5842 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦}) = {〈𝑥, 𝑦〉 ∣ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)} | |
5 | 3, 4 | eqtr4i 2766 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦} = ({〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦}) |
6 | relcoss 38405 | . . 3 ⊢ Rel ≀ ◡(𝐴 ⋉ 𝐵) | |
7 | dfrel4v 6212 | . . 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 38405 | . . . 4 ⊢ Rel ≀ ◡𝐴 | |
10 | dfrel4v 6212 | . . . 4 ⊢ (Rel ≀ ◡𝐴 ↔ ≀ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦}) | |
11 | 9, 10 | mpbi 230 | . . 3 ⊢ ≀ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦} |
12 | relcoss 38405 | . . . 4 ⊢ Rel ≀ ◡𝐵 | |
13 | dfrel4v 6212 | . . . 4 ⊢ (Rel ≀ ◡𝐵 ↔ ≀ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦}) | |
14 | 12, 13 | mpbi 230 | . . 3 ⊢ ≀ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦} |
15 | 11, 14 | ineq12i 4226 | . 2 ⊢ ( ≀ ◡𝐴 ∩ ≀ ◡𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦}) |
16 | 5, 8, 15 | 3eqtr4i 2773 | 1 ⊢ ≀ ◡(𝐴 ⋉ 𝐵) = ( ≀ ◡𝐴 ∩ ≀ ◡𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 Vcvv 3478 ∩ cin 3962 class class class wbr 5148 {copab 5210 ◡ccnv 5688 Rel wrel 5694 ⋉ cxrn 38161 ≀ ccoss 38162 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-1st 8013 df-2nd 8014 df-ec 8746 df-xrn 38353 df-coss 38393 |
This theorem is referenced by: disjxrn 38728 |
Copyright terms: Public domain | W3C validator |