Theorem ffnafv 31723

Description: A function maps to a class to which all values belong, analogous to ffnfv 6045. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
ffnafv	|- ( 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 ffnafv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffn 5729	. . 3 |- ( F : A --> B -> F Fn A )
2		fafvelrn 31722	. . . 4 |- ( ( F : A --> B /\ x e. A ) -> ( F''' x ) e. B )
3	2	ralrimiva 2878	. . 3 |- ( F : A --> B -> A. x e. A ( F''' x ) e. B )
4	1, 3	jca 532	. 2 |- ( F : A --> B -> ( F Fn A /\ A. x e. A ( F''' x ) e. B ) )
5		simpl 457	. . 3 |- ( ( F Fn A /\ A. x e. A ( F''' x ) e. B ) -> F Fn A )
6		afvelrnb0 31716	. . . . 5 |- ( F Fn A -> ( y e. ran F -> E. x e. A ( F''' x ) = y ) )
7		nfra1 2845	. . . . . 6 |- F/ x A. x e. A ( F''' x ) e. B
8		nfv 1683	. . . . . 6 |- F/ x y e. B
9		rsp 2830	. . . . . . 7 |- ( A. x e. A ( F''' x ) e. B -> ( x e. A -> ( F''' x ) e. B ) )
10		eleq1 2539	. . . . . . . 8 |- ( ( F''' x ) = y -> ( ( F''' x ) e. B <-> y e. B ) )
11	10	biimpcd 224	. . . . . . 7 |- ( ( F''' x ) e. B -> ( ( F''' x ) = y -> y e. B ) )
12	9, 11	syl6 33	. . . . . 6 |- ( A. x e. A ( F''' x ) e. B -> ( x e. A -> ( ( F''' x ) = y -> y e. B ) ) )
13	7, 8, 12	rexlimd 2947	. . . . 5 |- ( A. x e. A ( F''' x ) e. B -> ( E. x e. A ( F''' x ) = y -> y e. B ) )
14	6, 13	sylan9 657	. . . 4 |- ( ( F Fn A /\ A. x e. A ( F''' x ) e. B ) -> ( y e. ran F -> y e. B ) )
15	14	ssrdv 3510	. . 3 |- ( ( F Fn A /\ A. x e. A ( F''' x ) e. B ) -> ran F C_ B )
16		df-f 5590	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
17	5, 15, 16	sylanbrc 664	. 2 |- ( ( F Fn A /\ A. x e. A ( F''' x ) e. B ) -> F : A --> B )
18	4, 17	impbii 188	1 |- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F''' x ) e. B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   A.wral 2814   E.wrex 2815   C_ wss 3476   ran crn 5000   Fn wfn 5581   -->wf 5582  '''cafv 31666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-dfat 31668  df-afv 31669

This theorem is referenced by:  ffnaov  31751