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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 35729 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦 ↔ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦))) | |
2 | 1 | el2v 3501 | . . . 4 ⊢ (𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦 ↔ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)) |
3 | 2 | opabbii 5133 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦} = {〈𝑥, 𝑦〉 ∣ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)} |
4 | inopab 5701 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦}) = {〈𝑥, 𝑦〉 ∣ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)} | |
5 | 3, 4 | eqtr4i 2847 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦} = ({〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦}) |
6 | relcoss 35683 | . . 3 ⊢ Rel ≀ ◡(𝐴 ⋉ 𝐵) | |
7 | dfrel4v 6047 | . . 3 ⊢ (Rel ≀ ◡(𝐴 ⋉ 𝐵) ↔ ≀ ◡(𝐴 ⋉ 𝐵) = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦}) | |
8 | 6, 7 | mpbi 232 | . 2 ⊢ ≀ ◡(𝐴 ⋉ 𝐵) = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦} |
9 | relcoss 35683 | . . . 4 ⊢ Rel ≀ ◡𝐴 | |
10 | dfrel4v 6047 | . . . 4 ⊢ (Rel ≀ ◡𝐴 ↔ ≀ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦}) | |
11 | 9, 10 | mpbi 232 | . . 3 ⊢ ≀ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦} |
12 | relcoss 35683 | . . . 4 ⊢ Rel ≀ ◡𝐵 | |
13 | dfrel4v 6047 | . . . 4 ⊢ (Rel ≀ ◡𝐵 ↔ ≀ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦}) | |
14 | 12, 13 | mpbi 232 | . . 3 ⊢ ≀ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦} |
15 | 11, 14 | ineq12i 4187 | . 2 ⊢ ( ≀ ◡𝐴 ∩ ≀ ◡𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦}) |
16 | 5, 8, 15 | 3eqtr4i 2854 | 1 ⊢ ≀ ◡(𝐴 ⋉ 𝐵) = ( ≀ ◡𝐴 ∩ ≀ ◡𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 Vcvv 3494 ∩ cin 3935 class class class wbr 5066 {copab 5128 ◡ccnv 5554 Rel wrel 5560 ⋉ cxrn 35467 ≀ ccoss 35468 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fo 6361 df-fv 6363 df-1st 7689 df-2nd 7690 df-ec 8291 df-xrn 35638 df-coss 35674 |
This theorem is referenced by: disjxrn 35992 |
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