Theorem funfvima 5797

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 4968	. . . . . . 7 |- dom ( F |` A ) = ( A i^i dom F )
2	1	elin2 3352	. . . . . 6 |- ( B e. dom ( F |` A ) <-> ( B e. A /\ B e. dom F ) )
3		funres 5300	. . . . . . . . 9 |- ( Fun F -> Fun ( F |` A ) )
4		fvelrn 5696	. . . . . . . . 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 4677	. . . . . . . . . 10 |- ( F " A ) = ran ( F |` A )
7	6	eleq2i 2263	. . . . . . . . 9 |- ( ( F ` B ) e. ( F " A ) <-> ( F ` B ) e. ran ( F |` A ) )
8		fvres 5585	. . . . . . . . . 10 |- ( B e. A -> ( ( F |` A ) ` B ) = ( F ` B ) )
9	8	eleq1d 2265	. . . . . . . . 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 2167   dom cdm 4664   ran crn 4665   |` cres 4666   "cima 4667   Fun wfun 5253   ` cfv 5259

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267

This theorem is referenced by:  funfvima2  5798  fiintim  7001  caseinl  7166  caseinr  7167  ctssdccl  7186  suplocexprlemdisj  7804  suplocexprlemub  7807  ennnfonelemex  12656  ctinfomlemom  12669  txcnp  14591