Theorem opelf 5425

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 5421	. . . 4 |- ( F : A --> B -> F C_ ( A X. B ) )
2	1	sseld 3178	. . 3 |- ( F : A --> B -> ( <. C , D >. e. F -> <. C , D >. e. ( A X. B ) ) )
3		opelxp 4689	. . 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 2164   <.cop 3621   X. cxp 4657   -->wf 5250

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-dm 4669  df-rn 4670  df-fun 5256  df-fn 5257  df-f 5258

This theorem is referenced by:  feu  5436  fcnvres  5437  fsn  5730