Theorem ffnfv 5456

Description: A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)

Assertion

Ref	Expression
ffnfv	|- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )

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

Proof of Theorem ffnfv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffn 5161	. . 3 |- ( F : A --> B -> F Fn A )
2		ffvelrn 5432	. . . 4 |- ( ( F : A --> B /\ x e. A ) -> ( F ` x ) e. B )
3	2	ralrimiva 2446	. . 3 |- ( F : A --> B -> A. x e. A ( F ` x ) e. B )
4	1, 3	jca 300	. 2 |- ( F : A --> B -> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )
5		simpl 107	. . 3 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> F Fn A )
6		fvelrnb 5352	. . . . . 6 |- ( F Fn A -> ( y e. ran F <-> E. x e. A ( F ` x ) = y ) )
7	6	biimpd 142	. . . . 5 |- ( F Fn A -> ( y e. ran F -> E. x e. A ( F ` x ) = y ) )
8		nfra1 2409	. . . . . 6 |- F/ x A. x e. A ( F ` x ) e. B
9		nfv 1466	. . . . . 6 |- F/ x y e. B
10		rsp 2423	. . . . . . 7 |- ( A. x e. A ( F ` x ) e. B -> ( x e. A -> ( F ` x ) e. B ) )
11		eleq1 2150	. . . . . . . 8 |- ( ( F ` x ) = y -> ( ( F ` x ) e. B <-> y e. B ) )
12	11	biimpcd 157	. . . . . . 7 |- ( ( F ` x ) e. B -> ( ( F ` x ) = y -> y e. B ) )
13	10, 12	syl6 33	. . . . . 6 |- ( A. x e. A ( F ` x ) e. B -> ( x e. A -> ( ( F ` x ) = y -> y e. B ) ) )
14	8, 9, 13	rexlimd 2486	. . . . 5 |- ( A. x e. A ( F ` x ) e. B -> ( E. x e. A ( F ` x ) = y -> y e. B ) )
15	7, 14	sylan9 401	. . . 4 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> ( y e. ran F -> y e. B ) )
16	15	ssrdv 3031	. . 3 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> ran F C_ B )
17		df-f 5019	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
18	5, 16, 17	sylanbrc 408	. 2 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> F : A --> B )
19	4, 18	impbii 124	1 |- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   e. wcel 1438   A.wral 2359   E.wrex 2360   C_ wss 2999   ran crn 4439   Fn wfn 5010   -->wf 5011   ` cfv 5015

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023

This theorem is referenced by:  ffnfvf  5457  fnfvrnss  5458  fmpt2d  5460  ffnov  5749  cnref1o  9131  shftf  10260  eff2  10966  reeff1  10987