Theorem xpexg 2489

Description: The cross product of two sets is a set. Generalization of Proposition 6.2 of [TakeutiZaring] p. 23.

Assertion

Ref	Expression
xpexg	⊢ ((A ∈ C ∧ B ∈ D) → (A × B) ∈ V)

Proof of Theorem xpexg

Step	Hyp	Ref	Expression
1		xpeq1 2440	. . 3 ⊢ (x = A → (x × y) = (A × y))
2	1	eleq1d 1155	. 2 ⊢ (x = A → ((x × y) ∈ V ↔ (A × y) ∈ V))
3		xpeq2 2441	. . 3 ⊢ (y = B → (A × y) = (A × B))
4	3	eleq1d 1155	. 2 ⊢ (y = B → ((A × y) ∈ V ↔ (A × B) ∈ V))
5		visset 1350	. . 3 ⊢ x ∈ V
6		visset 1350	. . 3 ⊢ y ∈ V
7	5, 6	xpex 2488	. 2 ⊢ (x × y) ∈ V
8	2, 4, 7	vtocl2g 1386	1 ⊢ ((A ∈ C ∧ B ∈ D) → (A × B) ∈ V)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   × cxp 2408

This theorem is referenced by:  cnvexg 2669  coexg 2671  resfunexg 2717  fnex 2740  mapex 3261  cdavalt 3716

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-opab 2098  df-xp 2424  df-rel 2425

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