| Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > disjxrn | Structured version Visualization version GIF version | ||
| Description: Two ways of saying that a range Cartesian product is disjoint. (Contributed by Peter Mazsa, 17-Jun-2020.) (Revised by Peter Mazsa, 21-Sep-2021.) |
| Ref | Expression |
|---|---|
| disjxrn | ⊢ ( Disj (𝑅 ⋉ 𝑆) ↔ ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xrnrel 38374 | . . 3 ⊢ Rel (𝑅 ⋉ 𝑆) | |
| 2 | dfdisjALTV2 38715 | . . 3 ⊢ ( Disj (𝑅 ⋉ 𝑆) ↔ ( ≀ ◡(𝑅 ⋉ 𝑆) ⊆ I ∧ Rel (𝑅 ⋉ 𝑆))) | |
| 3 | 1, 2 | mpbiran2 710 | . 2 ⊢ ( Disj (𝑅 ⋉ 𝑆) ↔ ≀ ◡(𝑅 ⋉ 𝑆) ⊆ I ) |
| 4 | 1cosscnvxrn 38476 | . . 3 ⊢ ≀ ◡(𝑅 ⋉ 𝑆) = ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) | |
| 5 | 4 | sseq1i 4012 | . 2 ⊢ ( ≀ ◡(𝑅 ⋉ 𝑆) ⊆ I ↔ ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I ) |
| 6 | 3, 5 | bitri 275 | 1 ⊢ ( Disj (𝑅 ⋉ 𝑆) ↔ ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I ) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∩ cin 3950 ⊆ wss 3951 I cid 5577 ◡ccnv 5684 Rel wrel 5690 ⋉ cxrn 38181 ≀ ccoss 38182 Disj wdisjALTV 38216 |
| 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 df-cnvrefrel 38528 df-disjALTV 38706 |
| This theorem is referenced by: disjorimxrn 38749 |
| Copyright terms: Public domain | W3C validator |