Theorem disjsuc 39233

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 38788	. 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 6323	. . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
3	2	reseq2i 5935	. . . . 5 ⊢ (◡ E ↾ suc 𝐴) = (◡ E ↾ (𝐴 ∪ {𝐴}))
4	3	xrneq2i 38774	. . . 4 ⊢ (𝑅 ⋉ (◡ E ↾ suc 𝐴)) = (𝑅 ⋉ (◡ E ↾ (𝐴 ∪ {𝐴})))
5	4	disjeqi 39209	. . 3 ⊢ ( Disj (𝑅 ⋉ (◡ E ↾ suc 𝐴)) ↔ Disj (𝑅 ⋉ (◡ E ↾ (𝐴 ∪ {𝐴}))))
6		disjxrnres5 39221	. . 3 ⊢ ( Disj (𝑅 ⋉ (◡ E ↾ (𝐴 ∪ {𝐴}))) ↔ ∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅))
7	5, 6	bitri 276	. 2 ⊢ ( Disj (𝑅 ⋉ (◡ E ↾ suc 𝐴)) ↔ ∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅))
8		disjxrnres5 39221	. . 3 ⊢ ( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅))
9	8	anbi1i 630	. 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 315	1 ⊢ (𝐴 ∈ 𝑉 → ( Disj (𝑅 ⋉ (◡ E ↾ suc 𝐴)) ↔ ( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ ∀𝑢 ∈ 𝐴 ((𝑢 ∩ 𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119  ∀wral 3054   ∪ cun 3888   ∩ cin 3889  ∅c0 4268  {csn 4562   E cep 5524  ^◡ccnv 5624   ↾ cres 5627  suc csuc 6319  [cec 8638   ⋉ cxrn 38548   Disj wdisjALTV 38593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685  ax-reg 9504

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-1st 7938  df-2nd 7939  df-ec 8642  df-xrn 38754  df-coss 38875  df-cnvrefrel 38981  df-funALTV 39141  df-disjALTV 39164

This theorem is referenced by: (None)