Theorem ffvresb 5766

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 5451	. . . . . 6 |- ( ( F |` A ) : A --> B -> dom ( F |` A ) = A )
2		dmres 4999	. . . . . . 7 |- dom ( F |` A ) = ( A i^i dom F )
3		inss2 3402	. . . . . . 7 |- ( A i^i dom F ) C_ dom F
4	2, 3	eqsstri 3233	. . . . . 6 |- dom ( F |` A ) C_ dom F
5	1, 4	eqsstrrdi 3254	. . . . 5 |- ( ( F |` A ) : A --> B -> A C_ dom F )
6	5	sselda 3201	. . . 4 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> x e. dom F )
7		fvres 5623	. . . . . 6 |- ( x e. A -> ( ( F |` A ) ` x ) = ( F ` x ) )
8	7	adantl 277	. . . . 5 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> ( ( F |` A ) ` x ) = ( F ` x ) )
9		ffvelcdm 5736	. . . . 5 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> ( ( F |` A ) ` x ) e. B )
10	8, 9	eqeltrrd 2285	. . . 4 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> ( F ` x ) e. B )
11	6, 10	jca 306	. . 3 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> ( x e. dom F /\ ( F ` x ) e. B ) )
12	11	ralrimiva 2581	. 2 |- ( ( F |` A ) : A --> B -> A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) )
13		simpl 109	. . . . . . 7 |- ( ( x e. dom F /\ ( F ` x ) e. B ) -> x e. dom F )
14	13	ralimi 2571	. . . . . 6 |- ( A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) -> A. x e. A x e. dom F )
15		dfss3 3190	. . . . . 6 |- ( A C_ dom F <-> A. x e. A x e. dom F )
16	14, 15	sylibr 134	. . . . 5 |- ( A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) -> A C_ dom F )
17		funfn 5320	. . . . . 6 |- ( Fun F <-> F Fn dom F )
18		fnssres 5408	. . . . . 6 |- ( ( F Fn dom F /\ A C_ dom F ) -> ( F |` A ) Fn A )
19	17, 18	sylanb 284	. . . . 5 |- ( ( Fun F /\ A C_ dom F ) -> ( F |` A ) Fn A )
20	16, 19	sylan2 286	. . . 4 |- ( ( Fun F /\ A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) -> ( F |` A ) Fn A )
21		simpr 110	. . . . . . . 8 |- ( ( x e. dom F /\ ( F ` x ) e. B ) -> ( F ` x ) e. B )
22	7	eleq1d 2276	. . . . . . . 8 |- ( x e. A -> ( ( ( F |` A ) ` x ) e. B <-> ( F ` x ) e. B ) )
23	21, 22	imbitrrid 156	. . . . . . 7 |- ( x e. A -> ( ( x e. dom F /\ ( F ` x ) e. B ) -> ( ( F |` A ) ` x ) e. B ) )
24	23	ralimia 2569	. . . . . 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 277	. . . . 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 5763	. . . . 5 |- ( ( ( F |` A ) Fn A /\ A. x e. A ( ( F |` A ) ` x ) e. B ) -> ran ( F |` A ) C_ B )
27	20, 25, 26	syl2anc 411	. . . 4 |- ( ( Fun F /\ A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) -> ran ( F |` A ) C_ B )
28		df-f 5294	. . . 4 |- ( ( F |` A ) : A --> B <-> ( ( F |` A ) Fn A /\ ran ( F |` A ) C_ B ) )
29	20, 27, 28	sylanbrc 417	. . 3 |- ( ( Fun F /\ A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) -> ( F |` A ) : A --> B )
30	29	ex 115	. 2 |- ( Fun F -> ( A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) -> ( F |` A ) : A --> B ) )
31	12, 30	impbid2 143	1 |- ( Fun F -> ( ( F |` A ) : A --> B <-> A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   A.wral 2486   i^i cin 3173   C_ wss 3174   dom cdm 4693   ran crn 4694   |` cres 4695   Fun wfun 5284   Fn wfn 5285   -->wf 5286   ` cfv 5290

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298

This theorem is referenced by:  resflem  5767  tfrcl  6473  frecfcllem  6513  lmbr2  14801  lmff  14836