Theorem opelf 5535

Description: The members of an ordered pair element of a mapping belong to the mapping's domain and codomain. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
opelf	|- ( ( F : A --> B /\ <. C , D >. e. F ) -> ( C e. A /\ D e. B ) )

Proof of Theorem opelf

Step	Hyp	Ref	Expression
1		fssxp 5530	. . . 4 |- ( F : A --> B -> F C_ ( A X. B ) )
2	1	sseld 3237	. . 3 |- ( F : A --> B -> ( <. C , D >. e. F -> <. C , D >. e. ( A X. B ) ) )
3		opelxp 4779	. . 3 |- ( <. C , D >. e. ( A X. B ) <-> ( C e. A /\ D e. B ) )
4	2, 3	imbitrdi 161	. 2 |- ( F : A --> B -> ( <. C , D >. e. F -> ( C e. A /\ D e. B ) ) )
5	4	imp 124	1 |- ( ( F : A --> B /\ <. C , D >. e. F ) -> ( C e. A /\ D e. B ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2203   <.cop 3692   X. cxp 4747   -->wf 5348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-rel 4756  df-cnv 4757  df-dm 4759  df-rn 4760  df-fun 5354  df-fn 5355  df-f 5356

This theorem is referenced by:  feu  5549  fcnvres  5550  fsn  5849