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Theorem disjsuc 38141
Description: Disjoint range Cartesian product, special case. (Contributed by Peter Mazsa, 25-Aug-2023.)
Assertion
Ref Expression
disjsuc (𝐴𝑉 → ( Disj (𝑅 ⋉ ( E ↾ suc 𝐴)) ↔ ( Disj (𝑅 ⋉ ( E ↾ 𝐴)) ∧ ∀𝑢𝐴 ((𝑢𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))))
Distinct variable groups:   𝑢,𝐴   𝑢,𝑅   𝑢,𝑉

Proof of Theorem disjsuc
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 disjsuc2 37773 . 2 (𝐴𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅) ↔ (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅) ∧ ∀𝑢𝐴 ((𝑢𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))))
2 df-suc 6363 . . . . . 6 suc 𝐴 = (𝐴 ∪ {𝐴})
32reseq2i 5971 . . . . 5 ( E ↾ suc 𝐴) = ( E ↾ (𝐴 ∪ {𝐴}))
43xrneq2i 37763 . . . 4 (𝑅 ⋉ ( E ↾ suc 𝐴)) = (𝑅 ⋉ ( E ↾ (𝐴 ∪ {𝐴})))
54disjeqi 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 )) = ∅))
75, 6bitri 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)) ↔ ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅))
98anbi1i 623 . 2 (( Disj (𝑅 ⋉ ( E ↾ 𝐴)) ∧ ∀𝑢𝐴 ((𝑢𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)) ↔ (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅) ∧ ∀𝑢𝐴 ((𝑢𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))
101, 7, 93bitr4g 314 1 (𝐴𝑉 → ( Disj (𝑅 ⋉ ( E ↾ suc 𝐴)) ↔ ( Disj (𝑅 ⋉ ( E ↾ 𝐴)) ∧ ∀𝑢𝐴 ((𝑢𝐴) = ∅ ∨ ([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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