Theorem ffnfv 5687

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 5377	. . 3 |- ( F : A --> B -> F Fn A )
2		ffvelcdm 5662	. . . 4 |- ( ( F : A --> B /\ x e. A ) -> ( F ` x ) e. B )
3	2	ralrimiva 2560	. . 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 5576	. . . . . 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 2518	. . . . . 6 |- F/ x A. x e. A ( F ` x ) e. B
9		nfv 1538	. . . . . 6 |- F/ x y e. B
10		rsp 2534	. . . . . . 7 |- ( A. x e. A ( F ` x ) e. B -> ( x e. A -> ( F ` x ) e. B ) )
11		eleq1 2250	. . . . . . . 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 2601	. . . . 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 3173	. . 3 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> ran F C_ B )
17		df-f 5232	. . 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 1363   e. wcel 2158   A.wral 2465   E.wrex 2466   C_ wss 3141   ran crn 4639   Fn wfn 5223   -->wf 5224   ` cfv 5228

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236

This theorem is referenced by:  ffnfvf  5688  fnfvrnss  5689  fmpt2d  5691  ffnov  5992  elixpconst  6720  elixpsn  6749  ctssdccl  7124  cnref1o  9664  shftf  10853  eff2  11702  reeff1  11722  1arith  12379  ptex  12731  xpscf  12785  rngmgpf  13189  mgpf  13263  dvfre  14470  ioocosf1o  14571  012of  15042  2o01f  15043