Theorem fnovrn 6117

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 4725	. . 3 |- ( ( C e. A /\ D e. B ) -> <. C , D >. e. ( A X. B ) )
2		df-ov 5970	. . . 4 |- ( C F D ) = ( F ` <. C , D >. )
3		fnfvelrn 5735	. . . 4 |- ( ( F Fn ( A X. B ) /\ <. C , D >. e. ( A X. B ) ) -> ( F ` <. C , D >. ) e. ran F )
4	2, 3	eqeltrid 2294	. . 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 1202	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 981   e. wcel 2178   <.cop 3646   X. cxp 4691   ran crn 4694   Fn wfn 5285   ` cfv 5290  (class class class)co 5967

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-ov 5970

This theorem is referenced by:  unirnioo  10130  ioorebasg  10132  blelrnps  15006  blelrn  15007  xmettx  15097