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 35708 | . . . . 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 5125 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦} = {〈𝑥, 𝑦〉 ∣ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)} |
4 | inopab 5695 | . . 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 35662 | . . 3 ⊢ Rel ≀ ◡(𝐴 ⋉ 𝐵) | |
7 | dfrel4v 6041 | . . 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 35662 | . . . 4 ⊢ Rel ≀ ◡𝐴 | |
10 | dfrel4v 6041 | . . . 4 ⊢ (Rel ≀ ◡𝐴 ↔ ≀ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦}) | |
11 | 9, 10 | mpbi 232 | . . 3 ⊢ ≀ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐴𝑦} |
12 | relcoss 35662 | . . . 4 ⊢ Rel ≀ ◡𝐵 | |
13 | dfrel4v 6041 | . . . 4 ⊢ (Rel ≀ ◡𝐵 ↔ ≀ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦}) | |
14 | 12, 13 | mpbi 232 | . . 3 ⊢ ≀ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑥 ≀ ◡𝐵𝑦} |
15 | 11, 14 | ineq12i 4186 | . 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 1533 Vcvv 3494 ∩ cin 3934 class class class wbr 5058 {copab 5120 ◡ccnv 5548 Rel wrel 5554 ⋉ cxrn 35446 ≀ ccoss 35447 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fo 6355 df-fv 6357 df-1st 7683 df-2nd 7684 df-ec 8285 df-xrn 35617 df-coss 35653 |
This theorem is referenced by: disjxrn 35971 |
Copyright terms: Public domain | W3C validator |