Theorem opelf 2762

Description: The members of an ordered pair element of a mapping belong to the mapping's domain and codomain.

Hypothesis

Ref	Expression
opelf.1	⊢ D ∈ V

Assertion

Ref	Expression
opelf	⊢ ((F:A–→B ∧ ⟨C, D⟩ ∈ F) → (C ∈ A ∧ D ∈ B))

Proof of Theorem opelf

Step	Hyp	Ref	Expression
1		fssxp 2761	. . . 4 ⊢ (F:A–→B → F ⊆ (A × B))
2	1	sseld 1506	. . 3 ⊢ (F:A–→B → (⟨C, D⟩ ∈ F → ⟨C, D⟩ ∈ (A × B)))
3		opelf.1	. . . 4 ⊢ D ∈ V
4	3	opelxp 2452	. . 3 ⊢ (⟨C, D⟩ ∈ (A × B) ↔ (C ∈ A ∧ D ∈ B))
5	2, 4	syl6ib 185	. 2 ⊢ (F:A–→B → (⟨C, D⟩ ∈ F → (C ∈ A ∧ D ∈ B)))
6	5	imp 277	1 ⊢ ((F:A–→B ∧ ⟨C, D⟩ ∈ F) → (C ∈ A ∧ D ∈ B))

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   × cxp 2408  –→wf 2418

This theorem is referenced by:  fcoi2 2766  feu 2767  fcnvres 2768  fsn 2895

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  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434

metamath.org