Theorem ffvresb 5659

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 5353	. . . . . 6 |- ( ( F |` A ) : A --> B -> dom ( F |` A ) = A )
2		dmres 4912	. . . . . . 7 |- dom ( F |` A ) = ( A i^i dom F )
3		inss2 3348	. . . . . . 7 |- ( A i^i dom F ) C_ dom F
4	2, 3	eqsstri 3179	. . . . . 6 |- dom ( F |` A ) C_ dom F
5	1, 4	eqsstrrdi 3200	. . . . 5 |- ( ( F |` A ) : A --> B -> A C_ dom F )
6	5	sselda 3147	. . . 4 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> x e. dom F )
7		fvres 5520	. . . . . 6 |- ( x e. A -> ( ( F |` A ) ` x ) = ( F ` x ) )
8	7	adantl 275	. . . . 5 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> ( ( F |` A ) ` x ) = ( F ` x ) )
9		ffvelcdm 5629	. . . . 5 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> ( ( F |` A ) ` x ) e. B )
10	8, 9	eqeltrrd 2248	. . . 4 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> ( F ` x ) e. B )
11	6, 10	jca 304	. . 3 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> ( x e. dom F /\ ( F ` x ) e. B ) )
12	11	ralrimiva 2543	. 2 |- ( ( F |` A ) : A --> B -> A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) )
13		simpl 108	. . . . . . 7 |- ( ( x e. dom F /\ ( F ` x ) e. B ) -> x e. dom F )
14	13	ralimi 2533	. . . . . 6 |- ( A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) -> A. x e. A x e. dom F )
15		dfss3 3137	. . . . . 6 |- ( A C_ dom F <-> A. x e. A x e. dom F )
16	14, 15	sylibr 133	. . . . 5 |- ( A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) -> A C_ dom F )
17		funfn 5228	. . . . . 6 |- ( Fun F <-> F Fn dom F )
18		fnssres 5311	. . . . . 6 |- ( ( F Fn dom F /\ A C_ dom F ) -> ( F |` A ) Fn A )
19	17, 18	sylanb 282	. . . . 5 |- ( ( Fun F /\ A C_ dom F ) -> ( F |` A ) Fn A )
20	16, 19	sylan2 284	. . . 4 |- ( ( Fun F /\ A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) -> ( F |` A ) Fn A )
21		simpr 109	. . . . . . . 8 |- ( ( x e. dom F /\ ( F ` x ) e. B ) -> ( F ` x ) e. B )
22	7	eleq1d 2239	. . . . . . . 8 |- ( x e. A -> ( ( ( F |` A ) ` x ) e. B <-> ( F ` x ) e. B ) )
23	21, 22	syl5ibr 155	. . . . . . 7 |- ( x e. A -> ( ( x e. dom F /\ ( F ` x ) e. B ) -> ( ( F |` A ) ` x ) e. B ) )
24	23	ralimia 2531	. . . . . 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 275	. . . . 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 5656	. . . . 5 |- ( ( ( F |` A ) Fn A /\ A. x e. A ( ( F |` A ) ` x ) e. B ) -> ran ( F |` A ) C_ B )
27	20, 25, 26	syl2anc 409	. . . 4 |- ( ( Fun F /\ A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) -> ran ( F |` A ) C_ B )
28		df-f 5202	. . . 4 |- ( ( F |` A ) : A --> B <-> ( ( F |` A ) Fn A /\ ran ( F |` A ) C_ B ) )
29	20, 27, 28	sylanbrc 415	. . 3 |- ( ( Fun F /\ A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) -> ( F |` A ) : A --> B )
30	29	ex 114	. 2 |- ( Fun F -> ( A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) -> ( F |` A ) : A --> B ) )
31	12, 30	impbid2 142	1 |- ( Fun F -> ( ( F |` A ) : A --> B <-> A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   i^i cin 3120   C_ wss 3121   dom cdm 4611   ran crn 4612   |` cres 4613   Fun wfun 5192   Fn wfn 5193   -->wf 5194   ` cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206

This theorem is referenced by:  resflem  5660  tfrcl  6343  frecfcllem  6383  lmbr2  13008  lmff  13043