Theorem fnovrn 6022

Description: An operation's value belongs to its range. (Contributed by NM, 10-Feb-2007.)

Assertion

Ref	Expression
fnovrn	|- ( ( F Fn ( A X. B ) /\ C e. A /\ D e. B ) -> ( C F D ) e. ran F )

Proof of Theorem fnovrn

Step	Hyp	Ref	Expression
1		opelxpi 4659	. . 3 |- ( ( C e. A /\ D e. B ) -> <. C , D >. e. ( A X. B ) )
2		df-ov 5878	. . . 4 |- ( C F D ) = ( F ` <. C , D >. )
3		fnfvelrn 5649	. . . 4 |- ( ( F Fn ( A X. B ) /\ <. C , D >. e. ( A X. B ) ) -> ( F ` <. C , D >. ) e. ran F )
4	2, 3	eqeltrid 2264	. . 3 |- ( ( F Fn ( A X. B ) /\ <. C , D >. e. ( A X. B ) ) -> ( C F D ) e. ran F )
5	1, 4	sylan2 286	. 2 |- ( ( F Fn ( A X. B ) /\ ( C e. A /\ D e. B ) ) -> ( C F D ) e. ran F )
6	5	3impb 1199	1 |- ( ( F Fn ( A X. B ) /\ C e. A /\ D e. B ) -> ( C F D ) e. ran F )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   e. wcel 2148   <.cop 3596   X. cxp 4625   ran crn 4628   Fn wfn 5212   ` cfv 5217  (class class class)co 5875

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-fv 5225  df-ov 5878

This theorem is referenced by:  unirnioo  9973  ioorebasg  9975  blelrnps  13922  blelrn  13923  xmettx  14013