Theorem ffvresb 5349

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 5070	. . . . . 6 |- ( ( F |` A ) : A --> B -> dom ( F |` A ) = A )
2		dmres 4650	. . . . . . 7 |- dom ( F |` A ) = ( A i^i dom F )
3		inss2 3187	. . . . . . 7 |- ( A i^i dom F ) C_ dom F
4	2, 3	eqsstri 3029	. . . . . 6 |- dom ( F |` A ) C_ dom F
5	1, 4	syl6eqssr 3050	. . . . 5 |- ( ( F |` A ) : A --> B -> A C_ dom F )
6	5	sselda 2999	. . . 4 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> x e. dom F )
7		fvres 5219	. . . . . 6 |- ( x e. A -> ( ( F |` A ) ` x ) = ( F ` x ) )
8	7	adantl 271	. . . . 5 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> ( ( F |` A ) ` x ) = ( F ` x ) )
9		ffvelrn 5321	. . . . 5 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> ( ( F |` A ) ` x ) e. B )
10	8, 9	eqeltrrd 2156	. . . 4 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> ( F ` x ) e. B )
11	6, 10	jca 300	. . 3 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> ( x e. dom F /\ ( F ` x ) e. B ) )
12	11	ralrimiva 2434	. 2 |- ( ( F |` A ) : A --> B -> A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) )
13		simpl 107	. . . . . . 7 |- ( ( x e. dom F /\ ( F ` x ) e. B ) -> x e. dom F )
14	13	ralimi 2426	. . . . . 6 |- ( A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) -> A. x e. A x e. dom F )
15		dfss3 2989	. . . . . 6 |- ( A C_ dom F <-> A. x e. A x e. dom F )
16	14, 15	sylibr 132	. . . . 5 |- ( A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) -> A C_ dom F )
17		funfn 4951	. . . . . 6 |- ( Fun F <-> F Fn dom F )
18		fnssres 5032	. . . . . 6 |- ( ( F Fn dom F /\ A C_ dom F ) -> ( F |` A ) Fn A )
19	17, 18	sylanb 278	. . . . 5 |- ( ( Fun F /\ A C_ dom F ) -> ( F |` A ) Fn A )
20	16, 19	sylan2 280	. . . 4 |- ( ( Fun F /\ A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) -> ( F |` A ) Fn A )
21		simpr 108	. . . . . . . 8 |- ( ( x e. dom F /\ ( F ` x ) e. B ) -> ( F ` x ) e. B )
22	7	eleq1d 2147	. . . . . . . 8 |- ( x e. A -> ( ( ( F |` A ) ` x ) e. B <-> ( F ` x ) e. B ) )
23	21, 22	syl5ibr 154	. . . . . . 7 |- ( x e. A -> ( ( x e. dom F /\ ( F ` x ) e. B ) -> ( ( F |` A ) ` x ) e. B ) )
24	23	ralimia 2424	. . . . . 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 271	. . . . 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 5346	. . . . 5 |- ( ( ( F |` A ) Fn A /\ A. x e. A ( ( F |` A ) ` x ) e. B ) -> ran ( F |` A ) C_ B )
27	20, 25, 26	syl2anc 403	. . . 4 |- ( ( Fun F /\ A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) -> ran ( F |` A ) C_ B )
28		df-f 4926	. . . 4 |- ( ( F |` A ) : A --> B <-> ( ( F |` A ) Fn A /\ ran ( F |` A ) C_ B ) )
29	20, 27, 28	sylanbrc 408	. . 3 |- ( ( Fun F /\ A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) -> ( F |` A ) : A --> B )
30	29	ex 113	. 2 |- ( Fun F -> ( A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) -> ( F |` A ) : A --> B ) )
31	12, 30	impbid2 141	1 |- ( Fun F -> ( ( F |` A ) : A --> B <-> A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   i^i cin 2972   C_ wss 2973   dom cdm 4363   ran crn 4364   |` cres 4365   Fun wfun 4916   Fn wfn 4917   -->wf 4918   ` cfv 4922

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930

This theorem is referenced by: (None)