Theorem opelf 5359

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 5355	. . . 4 |- ( F : A --> B -> F C_ ( A X. B ) )
2	1	sseld 3141	. . 3 |- ( F : A --> B -> ( <. C , D >. e. F -> <. C , D >. e. ( A X. B ) ) )
3		opelxp 4634	. . 3 |- ( <. C , D >. e. ( A X. B ) <-> ( C e. A /\ D e. B ) )
4	2, 3	syl6ib 160	. 2 |- ( F : A --> B -> ( <. C , D >. e. F -> ( C e. A /\ D e. B ) ) )
5	4	imp 123	1 |- ( ( F : A --> B /\ <. C , D >. e. F ) -> ( C e. A /\ D e. B ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   <.cop 3579   X. cxp 4602   -->wf 5184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192

This theorem is referenced by:  feu  5370  fcnvres  5371  fsn  5657