Theorem ffnfv 5793

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 5473	. . 3 |- ( F : A --> B -> F Fn A )
2		ffvelcdm 5768	. . . 4 |- ( ( F : A --> B /\ x e. A ) -> ( F ` x ) e. B )
3	2	ralrimiva 2603	. . 3 |- ( F : A --> B -> A. x e. A ( F ` x ) e. B )
4	1, 3	jca 306	. 2 |- ( F : A --> B -> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )
5		simpl 109	. . 3 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> F Fn A )
6		fvelrnb 5681	. . . . . 6 |- ( F Fn A -> ( y e. ran F <-> E. x e. A ( F ` x ) = y ) )
7	6	biimpd 144	. . . . 5 |- ( F Fn A -> ( y e. ran F -> E. x e. A ( F ` x ) = y ) )
8		nfra1 2561	. . . . . 6 |- F/ x A. x e. A ( F ` x ) e. B
9		nfv 1574	. . . . . 6 |- F/ x y e. B
10		rsp 2577	. . . . . . 7 |- ( A. x e. A ( F ` x ) e. B -> ( x e. A -> ( F ` x ) e. B ) )
11		eleq1 2292	. . . . . . . 8 |- ( ( F ` x ) = y -> ( ( F ` x ) e. B <-> y e. B ) )
12	11	biimpcd 159	. . . . . . 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 2645	. . . . 5 |- ( A. x e. A ( F ` x ) e. B -> ( E. x e. A ( F ` x ) = y -> y e. B ) )
15	7, 14	sylan9 409	. . . 4 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> ( y e. ran F -> y e. B ) )
16	15	ssrdv 3230	. . 3 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> ran F C_ B )
17		df-f 5322	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
18	5, 16, 17	sylanbrc 417	. 2 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> F : A --> B )
19	4, 18	impbii 126	1 |- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   C_ wss 3197   ran crn 4720   Fn wfn 5313   -->wf 5314   ` cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326

This theorem is referenced by:  ffnfvf  5794  fnfvrnss  5795  fmpt2d  5797  ffnov  6108  elixpconst  6853  elixpsn  6882  ctssdccl  7278  cnref1o  9846  iswrdsymb  11089  ccatrn  11144  shftf  11341  eff2  12191  reeff1  12211  1arith  12890  ptex  13297  xpscf  13380  rngmgpf  13900  mgpf  13974  dvfre  15384  ioocosf1o  15528  012of  16357  2o01f  16358