![]() |
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 37238 | . . 3 ⊢ Rel (𝑅 ⋉ 𝑆) | |
2 | dfdisjALTV2 37579 | . . 3 ⊢ ( Disj (𝑅 ⋉ 𝑆) ↔ ( ≀ ◡(𝑅 ⋉ 𝑆) ⊆ I ∧ Rel (𝑅 ⋉ 𝑆))) | |
3 | 1, 2 | mpbiran2 708 | . 2 ⊢ ( Disj (𝑅 ⋉ 𝑆) ↔ ≀ ◡(𝑅 ⋉ 𝑆) ⊆ I ) |
4 | 1cosscnvxrn 37340 | . . 3 ⊢ ≀ ◡(𝑅 ⋉ 𝑆) = ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) | |
5 | 4 | sseq1i 4010 | . 2 ⊢ ( ≀ ◡(𝑅 ⋉ 𝑆) ⊆ I ↔ ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I ) |
6 | 3, 5 | bitri 274 | 1 ⊢ ( Disj (𝑅 ⋉ 𝑆) ↔ ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∩ cin 3947 ⊆ wss 3948 I cid 5573 ◡ccnv 5675 Rel wrel 5681 ⋉ cxrn 37037 ≀ ccoss 37038 Disj wdisjALTV 37072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-1st 7974 df-2nd 7975 df-ec 8704 df-xrn 37236 df-coss 37276 df-cnvrefrel 37392 df-disjALTV 37570 |
This theorem is referenced by: disjorimxrn 37613 |
Copyright terms: Public domain | W3C validator |