Theorem fnovrn 6202

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 4781	. . 3 |- ( ( C e. A /\ D e. B ) -> <. C , D >. e. ( A X. B ) )
2		df-ov 6053	. . . 4 |- ( C F D ) = ( F ` <. C , D >. )
3		fnfvelrn 5809	. . . 4 |- ( ( F Fn ( A X. B ) /\ <. C , D >. e. ( A X. B ) ) -> ( F ` <. C , D >. ) e. ran F )
4	2, 3	eqeltrid 2319	. . 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 2203   <.cop 3692   X. cxp 4747   ran crn 4750   Fn wfn 5347   ` cfv 5352  (class class class)co 6050

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-ov 6053

This theorem is referenced by:  unirnioo  10306  ioorebasg  10308  blelrnps  15284  blelrn  15285  xmettx  15375