Theorem opelxpg 2454

Description: Ordered pair membership in a cross product.

Assertion

Ref	Expression
opelxpg	⊢ (B ∈ R → (⟨A, B⟩ ∈ (C × D) ↔ (A ∈ C ∧ B ∈ D)))

Proof of Theorem opelxpg

Step	Hyp	Ref	Expression
1		opeq2 1877	. . 3 ⊢ (x = B → ⟨A, x⟩ = ⟨A, B⟩)
2	1	eleq1d 1155	. 2 ⊢ (x = B → (⟨A, x⟩ ∈ (C × D) ↔ ⟨A, B⟩ ∈ (C × D)))
3		eleq1 1149	. . 3 ⊢ (x = B → (x ∈ D ↔ B ∈ D))
4	3	anbi2d 468	. 2 ⊢ (x = B → ((A ∈ C ∧ x ∈ D) ↔ (A ∈ C ∧ B ∈ D)))
5		visset 1350	. . 3 ⊢ x ∈ V
6	5	opelxp 2452	. 2 ⊢ (⟨A, x⟩ ∈ (C × D) ↔ (A ∈ C ∧ x ∈ D))
7	2, 4, 6	vtoclbg 1384	1 ⊢ (B ∈ R → (⟨A, B⟩ ∈ (C × D) ↔ (A ∈ C ∧ B ∈ D)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   × cxp 2408

This theorem is referenced by:  opelxpi 2455  brelg 2458  ndmoprg 3057

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-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-opab 2098  df-xp 2424

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