Theorem fnovrn 6210

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 4786	. . 3 |- ( ( C e. A /\ D e. B ) -> <. C , D >. e. ( A X. B ) )
2		df-ov 6061	. . . 4 |- ( C F D ) = ( F ` <. C , D >. )
3		fnfvelrn 5814	. . . 4 |- ( ( F Fn ( A X. B ) /\ <. C , D >. e. ( A X. B ) ) -> ( F ` <. C , D >. ) e. ran F )
4	2, 3	eqeltrid 2321	. . 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 1226	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 1005   e. wcel 2205   <.cop 3697   X. cxp 4752   ran crn 4755   Fn wfn 5352   ` cfv 5357  (class class class)co 6058

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-ov 6061

This theorem is referenced by:  unirnioo  10325  ioorebasg  10327  blelrnps  15410  blelrn  15411  xmettx  15501