Theorem ffnfv 5740

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 5427	. . 3 |- ( F : A --> B -> F Fn A )
2		ffvelcdm 5715	. . . 4 |- ( ( F : A --> B /\ x e. A ) -> ( F ` x ) e. B )
3	2	ralrimiva 2579	. . 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 5628	. . . . . 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 2537	. . . . . 6 |- F/ x A. x e. A ( F ` x ) e. B
9		nfv 1551	. . . . . 6 |- F/ x y e. B
10		rsp 2553	. . . . . . 7 |- ( A. x e. A ( F ` x ) e. B -> ( x e. A -> ( F ` x ) e. B ) )
11		eleq1 2268	. . . . . . . 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 2620	. . . . 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 3199	. . 3 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> ran F C_ B )
17		df-f 5276	. . 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 1373   e. wcel 2176   A.wral 2484   E.wrex 2485   C_ wss 3166   ran crn 4677   Fn wfn 5267   -->wf 5268   ` cfv 5272

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 4163  ax-pow 4219  ax-pr 4254

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 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-fv 5280

This theorem is referenced by:  ffnfvf  5741  fnfvrnss  5742  fmpt2d  5744  ffnov  6051  elixpconst  6795  elixpsn  6824  ctssdccl  7215  cnref1o  9774  iswrdsymb  11014  ccatrn  11068  shftf  11174  eff2  12024  reeff1  12044  1arith  12723  ptex  13129  xpscf  13212  rngmgpf  13732  mgpf  13806  dvfre  15215  ioocosf1o  15359  012of  15967  2o01f  15968