Theorem ffvresb 5328

Description: A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)

Assertion

Ref	Expression
ffvresb	|- ( Fun F -> ( ( F |` A ) : A --> B <-> A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) )

Distinct variable groups:   x, A   x, B   x, F

Proof of Theorem ffvresb

Step	Hyp	Ref	Expression
1		fdm 5050	. . . . . 6 |- ( ( F |` A ) : A --> B -> dom ( F |` A ) = A )
2		dmres 4632	. . . . . . 7 |- dom ( F |` A ) = ( A i^i dom F )
3		inss2 3158	. . . . . . 7 |- ( A i^i dom F ) C_ dom F
4	2, 3	eqsstri 2975	. . . . . 6 |- dom ( F |` A ) C_ dom F
5	1, 4	syl6eqssr 2996	. . . . 5 |- ( ( F |` A ) : A --> B -> A C_ dom F )
6	5	sselda 2945	. . . 4 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> x e. dom F )
7		fvres 5198	. . . . . 6 |- ( x e. A -> ( ( F |` A ) ` x ) = ( F ` x ) )
8	7	adantl 262	. . . . 5 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> ( ( F |` A ) ` x ) = ( F ` x ) )
9		ffvelrn 5300	. . . . 5 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> ( ( F |` A ) ` x ) e. B )
10	8, 9	eqeltrrd 2115	. . . 4 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> ( F ` x ) e. B )
11	6, 10	jca 290	. . 3 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> ( x e. dom F /\ ( F ` x ) e. B ) )
12	11	ralrimiva 2392	. 2 |- ( ( F |` A ) : A --> B -> A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) )
13		simpl 102	. . . . . . 7 |- ( ( x e. dom F /\ ( F ` x ) e. B ) -> x e. dom F )
14	13	ralimi 2384	. . . . . 6 |- ( A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) -> A. x e. A x e. dom F )
15		dfss3 2935	. . . . . 6 |- ( A C_ dom F <-> A. x e. A x e. dom F )
16	14, 15	sylibr 137	. . . . 5 |- ( A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) -> A C_ dom F )
17		funfn 4931	. . . . . 6 |- ( Fun F <-> F Fn dom F )
18		fnssres 5012	. . . . . 6 |- ( ( F Fn dom F /\ A C_ dom F ) -> ( F |` A ) Fn A )
19	17, 18	sylanb 268	. . . . 5 |- ( ( Fun F /\ A C_ dom F ) -> ( F |` A ) Fn A )
20	16, 19	sylan2 270	. . . 4 |- ( ( Fun F /\ A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) -> ( F |` A ) Fn A )
21		simpr 103	. . . . . . . 8 |- ( ( x e. dom F /\ ( F ` x ) e. B ) -> ( F ` x ) e. B )
22	7	eleq1d 2106	. . . . . . . 8 |- ( x e. A -> ( ( ( F |` A ) ` x ) e. B <-> ( F ` x ) e. B ) )
23	21, 22	syl5ibr 145	. . . . . . 7 |- ( x e. A -> ( ( x e. dom F /\ ( F ` x ) e. B ) -> ( ( F |` A ) ` x ) e. B ) )
24	23	ralimia 2382	. . . . . 6 |- ( A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) -> A. x e. A ( ( F |` A ) ` x ) e. B )
25	24	adantl 262	. . . . 5 |- ( ( Fun F /\ A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) -> A. x e. A ( ( F |` A ) ` x ) e. B )
26		fnfvrnss 5325	. . . . 5 |- ( ( ( F |` A ) Fn A /\ A. x e. A ( ( F |` A ) ` x ) e. B ) -> ran ( F |` A ) C_ B )
27	20, 25, 26	syl2anc 391	. . . 4 |- ( ( Fun F /\ A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) -> ran ( F |` A ) C_ B )
28		df-f 4906	. . . 4 |- ( ( F |` A ) : A --> B <-> ( ( F |` A ) Fn A /\ ran ( F |` A ) C_ B ) )
29	20, 27, 28	sylanbrc 394	. . 3 |- ( ( Fun F /\ A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) -> ( F |` A ) : A --> B )
30	29	ex 108	. 2 |- ( Fun F -> ( A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) -> ( F |` A ) : A --> B ) )
31	12, 30	impbid2 131	1 |- ( Fun F -> ( ( F |` A ) : A --> B <-> A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   A.wral 2306   i^i cin 2916   C_ wss 2917   dom cdm 4345   ran crn 4346   |` cres 4347   Fun wfun 4896   Fn wfn 4897   -->wf 4898   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910

This theorem is referenced by: (None)