Theorem ffnfv 5586

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 5280	. . 3 |- ( F : A --> B -> F Fn A )
2		ffvelrn 5561	. . . 4 |- ( ( F : A --> B /\ x e. A ) -> ( F ` x ) e. B )
3	2	ralrimiva 2508	. . 3 |- ( F : A --> B -> A. x e. A ( F ` x ) e. B )
4	1, 3	jca 304	. 2 |- ( F : A --> B -> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )
5		simpl 108	. . 3 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> F Fn A )
6		fvelrnb 5477	. . . . . 6 |- ( F Fn A -> ( y e. ran F <-> E. x e. A ( F ` x ) = y ) )
7	6	biimpd 143	. . . . 5 |- ( F Fn A -> ( y e. ran F -> E. x e. A ( F ` x ) = y ) )
8		nfra1 2469	. . . . . 6 |- F/ x A. x e. A ( F ` x ) e. B
9		nfv 1509	. . . . . 6 |- F/ x y e. B
10		rsp 2483	. . . . . . 7 |- ( A. x e. A ( F ` x ) e. B -> ( x e. A -> ( F ` x ) e. B ) )
11		eleq1 2203	. . . . . . . 8 |- ( ( F ` x ) = y -> ( ( F ` x ) e. B <-> y e. B ) )
12	11	biimpcd 158	. . . . . . 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 2549	. . . . 5 |- ( A. x e. A ( F ` x ) e. B -> ( E. x e. A ( F ` x ) = y -> y e. B ) )
15	7, 14	sylan9 407	. . . 4 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> ( y e. ran F -> y e. B ) )
16	15	ssrdv 3108	. . 3 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> ran F C_ B )
17		df-f 5135	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
18	5, 16, 17	sylanbrc 414	. 2 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> F : A --> B )
19	4, 18	impbii 125	1 |- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   A.wral 2417   E.wrex 2418   C_ wss 3076   ran crn 4548   Fn wfn 5126   -->wf 5127   ` cfv 5131

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139

This theorem is referenced by:  ffnfvf  5587  fnfvrnss  5588  fmpt2d  5590  ffnov  5883  elixpconst  6608  elixpsn  6637  ctssdccl  7004  cnref1o  9469  shftf  10634  eff2  11423  reeff1  11443  dvfre  12882  ioocosf1o  12983  012of  13363  2o01f  13364