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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjsuc | Structured version Visualization version GIF version |
Description: Disjoint range Cartesian product, special case. (Contributed by Peter Mazsa, 25-Aug-2023.) |
Ref | Expression |
---|---|
disjsuc | ⊢ (𝐴 ∈ 𝑉 → ( Disj (𝑅 ⋉ (◡ E ↾ suc 𝐴)) ↔ ( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ ∀𝑢 ∈ 𝐴 ((𝑢 ∩ 𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjsuc2 37773 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅) ↔ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅) ∧ ∀𝑢 ∈ 𝐴 ((𝑢 ∩ 𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))) | |
2 | df-suc 6363 | . . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
3 | 2 | reseq2i 5971 | . . . . 5 ⊢ (◡ E ↾ suc 𝐴) = (◡ E ↾ (𝐴 ∪ {𝐴})) |
4 | 3 | xrneq2i 37763 | . . . 4 ⊢ (𝑅 ⋉ (◡ E ↾ suc 𝐴)) = (𝑅 ⋉ (◡ E ↾ (𝐴 ∪ {𝐴}))) |
5 | 4 | disjeqi 38117 | . . 3 ⊢ ( Disj (𝑅 ⋉ (◡ E ↾ suc 𝐴)) ↔ Disj (𝑅 ⋉ (◡ E ↾ (𝐴 ∪ {𝐴})))) |
6 | disjxrnres5 38129 | . . 3 ⊢ ( Disj (𝑅 ⋉ (◡ E ↾ (𝐴 ∪ {𝐴}))) ↔ ∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅)) | |
7 | 5, 6 | bitri 275 | . 2 ⊢ ( Disj (𝑅 ⋉ (◡ E ↾ suc 𝐴)) ↔ ∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅)) |
8 | disjxrnres5 38129 | . . 3 ⊢ ( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅)) | |
9 | 8 | anbi1i 623 | . 2 ⊢ (( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ ∀𝑢 ∈ 𝐴 ((𝑢 ∩ 𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)) ↔ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅) ∧ ∀𝑢 ∈ 𝐴 ((𝑢 ∩ 𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))) |
10 | 1, 7, 9 | 3bitr4g 314 | 1 ⊢ (𝐴 ∈ 𝑉 → ( Disj (𝑅 ⋉ (◡ E ↾ suc 𝐴)) ↔ ( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ ∀𝑢 ∈ 𝐴 ((𝑢 ∩ 𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∪ cun 3941 ∩ cin 3942 ∅c0 4317 {csn 4623 E cep 5572 ◡ccnv 5668 ↾ cres 5671 suc csuc 6359 [cec 8700 ⋉ cxrn 37554 Disj wdisjALTV 37589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 ax-reg 9586 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-eprel 5573 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fo 6542 df-fv 6544 df-1st 7971 df-2nd 7972 df-ec 8704 df-xrn 37753 df-coss 37793 df-cnvrefrel 37909 df-funALTV 38064 df-disjALTV 38087 |
This theorem is referenced by: (None) |
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