Theorem funfvima 5750

Description: A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.)

Assertion

Ref	Expression
funfvima	|- ( ( Fun F /\ B e. dom F ) -> ( B e. A -> ( F ` B ) e. ( F " A ) ) )

Proof of Theorem funfvima

Step	Hyp	Ref	Expression
1		dmres 4930	. . . . . . 7 |- dom ( F |` A ) = ( A i^i dom F )
2	1	elin2 3325	. . . . . 6 |- ( B e. dom ( F |` A ) <-> ( B e. A /\ B e. dom F ) )
3		funres 5259	. . . . . . . . 9 |- ( Fun F -> Fun ( F |` A ) )
4		fvelrn 5649	. . . . . . . . 9 |- ( ( Fun ( F |` A ) /\ B e. dom ( F |` A ) ) -> ( ( F |` A ) ` B ) e. ran ( F |` A ) )
5	3, 4	sylan 283	. . . . . . . 8 |- ( ( Fun F /\ B e. dom ( F |` A ) ) -> ( ( F |` A ) ` B ) e. ran ( F |` A ) )
6		df-ima 4641	. . . . . . . . . 10 |- ( F " A ) = ran ( F |` A )
7	6	eleq2i 2244	. . . . . . . . 9 |- ( ( F ` B ) e. ( F " A ) <-> ( F ` B ) e. ran ( F |` A ) )
8		fvres 5541	. . . . . . . . . 10 |- ( B e. A -> ( ( F |` A ) ` B ) = ( F ` B ) )
9	8	eleq1d 2246	. . . . . . . . 9 |- ( B e. A -> ( ( ( F |` A ) ` B ) e. ran ( F |` A ) <-> ( F ` B ) e. ran ( F |` A ) ) )
10	7, 9	bitr4id 199	. . . . . . . 8 |- ( B e. A -> ( ( F ` B ) e. ( F " A ) <-> ( ( F |` A ) ` B ) e. ran ( F |` A ) ) )
11	5, 10	syl5ibrcom 157	. . . . . . 7 |- ( ( Fun F /\ B e. dom ( F |` A ) ) -> ( B e. A -> ( F ` B ) e. ( F " A ) ) )
12	11	ex 115	. . . . . 6 |- ( Fun F -> ( B e. dom ( F |` A ) -> ( B e. A -> ( F ` B ) e. ( F " A ) ) ) )
13	2, 12	biimtrrid 153	. . . . 5 |- ( Fun F -> ( ( B e. A /\ B e. dom F ) -> ( B e. A -> ( F ` B ) e. ( F " A ) ) ) )
14	13	expd 258	. . . 4 |- ( Fun F -> ( B e. A -> ( B e. dom F -> ( B e. A -> ( F ` B ) e. ( F " A ) ) ) ) )
15	14	com12 30	. . 3 |- ( B e. A -> ( Fun F -> ( B e. dom F -> ( B e. A -> ( F ` B ) e. ( F " A ) ) ) ) )
16	15	impd 254	. 2 |- ( B e. A -> ( ( Fun F /\ B e. dom F ) -> ( B e. A -> ( F ` B ) e. ( F " A ) ) ) )
17	16	pm2.43b 52	1 |- ( ( Fun F /\ B e. dom F ) -> ( B e. A -> ( F ` B ) e. ( F " A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2148   dom cdm 4628   ran crn 4629   |` cres 4630   "cima 4631   Fun wfun 5212   ` cfv 5218

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 4123  ax-pow 4176  ax-pr 4211

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 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226

This theorem is referenced by:  funfvima2  5751  fiintim  6930  caseinl  7092  caseinr  7093  ctssdccl  7112  suplocexprlemdisj  7721  suplocexprlemub  7724  ennnfonelemex  12417  ctinfomlemom  12430  txcnp  13810