Theorem funfvima 5870

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 5025	. . . . . . 7 |- dom ( F |` A ) = ( A i^i dom F )
2	1	elin2 3392	. . . . . 6 |- ( B e. dom ( F |` A ) <-> ( B e. A /\ B e. dom F ) )
3		funres 5358	. . . . . . . . 9 |- ( Fun F -> Fun ( F |` A ) )
4		fvelrn 5765	. . . . . . . . 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 4731	. . . . . . . . . 10 |- ( F " A ) = ran ( F |` A )
7	6	eleq2i 2296	. . . . . . . . 9 |- ( ( F ` B ) e. ( F " A ) <-> ( F ` B ) e. ran ( F |` A ) )
8		fvres 5650	. . . . . . . . . 10 |- ( B e. A -> ( ( F |` A ) ` B ) = ( F ` B ) )
9	8	eleq1d 2298	. . . . . . . . 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 2200   dom cdm 4718   ran crn 4719   |` cres 4720   "cima 4721   Fun wfun 5311   ` cfv 5317

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325

This theorem is referenced by:  funfvima2  5871  elovimad  6044  fiintim  7089  caseinl  7254  caseinr  7255  ctssdccl  7274  suplocexprlemdisj  7903  suplocexprlemub  7906  swrdwrdsymbg  11191  ennnfonelemex  12980  ctinfomlemom  12993  txcnp  14939