Theorem disjsuc 39885

Description: Disjoint range Cartesian product, special case. (Contributed by Peter Mazsa, 25-Aug-2023.)

Assertion

Ref	Expression
disjsuc	⊢ (𝐴 ∈ 𝑉 → ( Disj (𝑅 ⋉ (◡ E ↾ suc 𝐴)) ↔ ( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ ∀𝑢 ∈ 𝐴 ((𝑢 ∩ 𝐴) = ∅ ∨ ([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.

Step	Hyp	Ref	Expression
1		disjsuc2 39440	. 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 6368	. . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
3	2	reseq2i 5966	. . . . 5 ⊢ (◡ E ↾ suc 𝐴) = (◡ E ↾ (𝐴 ∪ {𝐴}))
4	3	xrneq2i 39426	. . . 4 ⊢ (𝑅 ⋉ (◡ E ↾ suc 𝐴)) = (𝑅 ⋉ (◡ E ↾ (𝐴 ∪ {𝐴})))
5	4	disjeqi 39861	. . 3 ⊢ ( Disj (𝑅 ⋉ (◡ E ↾ suc 𝐴)) ↔ Disj (𝑅 ⋉ (◡ E ↾ (𝐴 ∪ {𝐴}))))
6		disjxrnres5 39873	. . 3 ⊢ ( Disj (𝑅 ⋉ (◡ E ↾ (𝐴 ∪ {𝐴}))) ↔ ∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅))
7	5, 6	bitri 278	. 2 ⊢ ( Disj (𝑅 ⋉ (◡ E ↾ suc 𝐴)) ↔ ∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅))
8		disjxrnres5 39873	. . 3 ⊢ ( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅))
9	8	anbi1i 636	. 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 317	1 ⊢ (𝐴 ∈ 𝑉 → ( Disj (𝑅 ⋉ (◡ E ↾ suc 𝐴)) ↔ ( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ ∀𝑢 ∈ 𝐴 ((𝑢 ∩ 𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ∩ cin 4089   ∪ cun 4149  ∅c0 4278  {csn 4584   E cep 5549  ^◡ccnv 5649   ↾ cres 5652  suc csuc 6364  [cec 8719   ⋉ cxrn 39200   Disj wdisjALTV 39245

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7752  ax-reg 9590

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-eprel 5550  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fo 6545  df-fv 6547  df-1st 8002  df-2nd 8003  df-ec 8723  df-xrn 39406  df-coss 39527  df-cnvrefrel 39633  df-funALTV 39793  df-disjALTV 39816

This theorem is used by: (None)