Theorem brelg 2458

Description: Two things in a binary relation belong to the relation's domain.

Hypothesis

Ref	Expression
brelg.1	⊢ R ⊆ (C × D)

Assertion

Ref	Expression
brelg	⊢ (B ∈ S → (ARB → (A ∈ C ∧ B ∈ D)))

Proof of Theorem brelg

Step	Hyp	Ref	Expression
1		opelxpg 2454	. . 3 ⊢ (B ∈ S → (⟨A, B⟩ ∈ (C × D) ↔ (A ∈ C ∧ B ∈ D)))
2		brelg.1	. . . 4 ⊢ R ⊆ (C × D)
3	2	sseli 1504	. . 3 ⊢ (⟨A, B⟩ ∈ R → ⟨A, B⟩ ∈ (C × D))
4	1, 3	syl5bi 183	. 2 ⊢ (B ∈ S → (⟨A, B⟩ ∈ R → (A ∈ C ∧ B ∈ D)))
5		df-br 2063	. 2 ⊢ (ARB ↔ ⟨A, B⟩ ∈ R)
6	4, 5	syl5ib 181	1 ⊢ (B ∈ S → (ARB → (A ∈ C ∧ B ∈ D)))

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092   ⊆ wss 1487  ⟨cop 1810   class class class wbr 2054   × cxp 2408

This theorem is referenced by:  brel 2459  suplem2pr 3956

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-br 2063  df-opab 2098  df-xp 2424

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