Theorem opelf 5507

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 5502	. . . 4 |- ( F : A --> B -> F C_ ( A X. B ) )
2	1	sseld 3226	. . 3 |- ( F : A --> B -> ( <. C , D >. e. F -> <. C , D >. e. ( A X. B ) ) )
3		opelxp 4755	. . 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 2202   <.cop 3672   X. cxp 4723   -->wf 5322

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736  df-fun 5328  df-fn 5329  df-f 5330

This theorem is referenced by:  feu  5519  fcnvres  5520  fsn  5819