Theorem funfvima 5526

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 4734	. . . . . . 7 |- dom ( F |` A ) = ( A i^i dom F )
2	1	elin2 3188	. . . . . 6 |- ( B e. dom ( F |` A ) <-> ( B e. A /\ B e. dom F ) )
3		funres 5055	. . . . . . . . 9 |- ( Fun F -> Fun ( F |` A ) )
4		fvelrn 5430	. . . . . . . . 9 |- ( ( Fun ( F |` A ) /\ B e. dom ( F |` A ) ) -> ( ( F |` A ) ` B ) e. ran ( F |` A ) )
5	3, 4	sylan 277	. . . . . . . 8 |- ( ( Fun F /\ B e. dom ( F |` A ) ) -> ( ( F |` A ) ` B ) e. ran ( F |` A ) )
6		fvres 5329	. . . . . . . . . 10 |- ( B e. A -> ( ( F |` A ) ` B ) = ( F ` B ) )
7	6	eleq1d 2156	. . . . . . . . 9 |- ( B e. A -> ( ( ( F |` A ) ` B ) e. ran ( F |` A ) <-> ( F ` B ) e. ran ( F |` A ) ) )
8		df-ima 4451	. . . . . . . . . 10 |- ( F " A ) = ran ( F |` A )
9	8	eleq2i 2154	. . . . . . . . 9 |- ( ( F ` B ) e. ( F " A ) <-> ( F ` B ) e. ran ( F |` A ) )
10	7, 9	syl6rbbr 197	. . . . . . . 8 |- ( B e. A -> ( ( F ` B ) e. ( F " A ) <-> ( ( F |` A ) ` B ) e. ran ( F |` A ) ) )
11	5, 10	syl5ibrcom 155	. . . . . . 7 |- ( ( Fun F /\ B e. dom ( F |` A ) ) -> ( B e. A -> ( F ` B ) e. ( F " A ) ) )
12	11	ex 113	. . . . . 6 |- ( Fun F -> ( B e. dom ( F |` A ) -> ( B e. A -> ( F ` B ) e. ( F " A ) ) ) )
13	2, 12	syl5bir 151	. . . . 5 |- ( Fun F -> ( ( B e. A /\ B e. dom F ) -> ( B e. A -> ( F ` B ) e. ( F " A ) ) ) )
14	13	expd 254	. . . 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 251	. 2 |- ( B e. A -> ( ( Fun F /\ B e. dom F ) -> ( B e. A -> ( F ` B ) e. ( F " A ) ) ) )
17	16	pm2.43b 51	1 |- ( ( Fun F /\ B e. dom F ) -> ( B e. A -> ( F ` B ) e. ( F " A ) ) )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1438   dom cdm 4438   ran crn 4439   |` cres 4440   "cima 4441   Fun wfun 5009   ` cfv 5015

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023

This theorem is referenced by:  funfvima2  5527  fiintim  6637  djuun  6758