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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjxrnres5 | Structured version Visualization version GIF version |
Description: Disjoint range Cartesian product. (Contributed by Peter Mazsa, 25-Aug-2023.) |
Ref | Expression |
---|---|
disjxrnres5 | ⊢ ( Disj (𝑅 ⋉ (𝑆 ↾ 𝐴)) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ 𝑆) ∩ [𝑣](𝑅 ⋉ 𝑆)) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrnres2 37179 | . . 3 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴)) | |
2 | 1 | disjeqi 37511 | . 2 ⊢ ( Disj ((𝑅 ⋉ 𝑆) ↾ 𝐴) ↔ Disj (𝑅 ⋉ (𝑆 ↾ 𝐴))) |
3 | xrnrel 37149 | . . 3 ⊢ Rel (𝑅 ⋉ 𝑆) | |
4 | disjres 37520 | . . 3 ⊢ (Rel (𝑅 ⋉ 𝑆) → ( Disj ((𝑅 ⋉ 𝑆) ↾ 𝐴) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ 𝑆) ∩ [𝑣](𝑅 ⋉ 𝑆)) = ∅))) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ ( Disj ((𝑅 ⋉ 𝑆) ↾ 𝐴) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ 𝑆) ∩ [𝑣](𝑅 ⋉ 𝑆)) = ∅)) |
6 | 2, 5 | bitr3i 277 | 1 ⊢ ( Disj (𝑅 ⋉ (𝑆 ↾ 𝐴)) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ 𝑆) ∩ [𝑣](𝑅 ⋉ 𝑆)) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 846 = wceq 1542 ∀wral 3062 ∩ cin 3945 ∅c0 4320 ↾ cres 5674 Rel wrel 5677 [cec 8689 ⋉ cxrn 36948 Disj wdisjALTV 36983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rmo 3377 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-opab 5207 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ec 8693 df-xrn 37147 df-coss 37187 df-cnvrefrel 37303 df-funALTV 37458 df-disjALTV 37481 |
This theorem is referenced by: disjsuc 37535 |
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