Theorem funfvima 5885

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 5034	. . . . . . 7 |- dom ( F |` A ) = ( A i^i dom F )
2	1	elin2 3395	. . . . . 6 |- ( B e. dom ( F |` A ) <-> ( B e. A /\ B e. dom F ) )
3		funres 5367	. . . . . . . . 9 |- ( Fun F -> Fun ( F |` A ) )
4		fvelrn 5778	. . . . . . . . 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 4738	. . . . . . . . . 10 |- ( F " A ) = ran ( F |` A )
7	6	eleq2i 2298	. . . . . . . . 9 |- ( ( F ` B ) e. ( F " A ) <-> ( F ` B ) e. ran ( F |` A ) )
8		fvres 5663	. . . . . . . . . 10 |- ( B e. A -> ( ( F |` A ) ` B ) = ( F ` B ) )
9	8	eleq1d 2300	. . . . . . . . 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 2202   dom cdm 4725   ran crn 4726   |` cres 4727   "cima 4728   Fun wfun 5320   ` cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334

This theorem is referenced by:  funfvima2  5886  elovimad  6061  fiintim  7122  caseinl  7289  caseinr  7290  ctssdccl  7309  suplocexprlemdisj  7939  suplocexprlemub  7942  swrdwrdsymbg  11244  ennnfonelemex  13034  ctinfomlemom  13047  txcnp  14994