Theorem ffvresb 5743

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 5431	. . . . . 6 |- ( ( F |` A ) : A --> B -> dom ( F |` A ) = A )
2		dmres 4980	. . . . . . 7 |- dom ( F |` A ) = ( A i^i dom F )
3		inss2 3394	. . . . . . 7 |- ( A i^i dom F ) C_ dom F
4	2, 3	eqsstri 3225	. . . . . 6 |- dom ( F |` A ) C_ dom F
5	1, 4	eqsstrrdi 3246	. . . . 5 |- ( ( F |` A ) : A --> B -> A C_ dom F )
6	5	sselda 3193	. . . 4 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> x e. dom F )
7		fvres 5600	. . . . . 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 5713	. . . . 5 |- ( ( ( F |` A ) : A --> B /\ x e. A ) -> ( ( F |` A ) ` x ) e. B )
10	8, 9	eqeltrrd 2283	. . . 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 2579	. 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 2569	. . . . . 6 |- ( A. x e. A ( x e. dom F /\ ( F ` x ) e. B ) -> A. x e. A x e. dom F )
15		dfss3 3182	. . . . . 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 5301	. . . . . 6 |- ( Fun F <-> F Fn dom F )
18		fnssres 5389	. . . . . 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 2274	. . . . . . . 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 2567	. . . . . 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 5740	. . . . 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 5275	. . . 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 2176   A.wral 2484   i^i cin 3165   C_ wss 3166   dom cdm 4675   ran crn 4676   |` cres 4677   Fun wfun 5265   Fn wfn 5266   -->wf 5267   ` cfv 5271

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279

This theorem is referenced by:  resflem  5744  tfrcl  6450  frecfcllem  6490  lmbr2  14686  lmff  14721