Theorem ffnafv 38763

Description: A function maps to a class to which all values belong, analogous to ffnfv 6032. (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 5710	. . 3 |- ( F : A --> B -> F Fn A )
2		fafvelrn 38762	. . . 4 |- ( ( F : A --> B /\ x e. A ) -> ( F''' x ) e. B )
3	2	ralrimiva 2789	. . 3 |- ( F : A --> B -> A. x e. A ( F''' x ) e. B )
4	1, 3	jca 539	. 2 |- ( F : A --> B -> ( F Fn A /\ A. x e. A ( F''' x ) e. B ) )
5		simpl 463	. . 3 |- ( ( F Fn A /\ A. x e. A ( F''' x ) e. B ) -> F Fn A )
6		afvelrnb0 38756	. . . . 5 |- ( F Fn A -> ( y e. ran F -> E. x e. A ( F''' x ) = y ) )
7		nfra1 2765	. . . . . 6 |- F/ x A. x e. A ( F''' x ) e. B
8		nfv 1764	. . . . . 6 |- F/ x y e. B
9		rsp 2753	. . . . . . 7 |- ( A. x e. A ( F''' x ) e. B -> ( x e. A -> ( F''' x ) e. B ) )
10		eleq1 2517	. . . . . . . 8 |- ( ( F''' x ) = y -> ( ( F''' x ) e. B <-> y e. B ) )
11	10	biimpcd 232	. . . . . . 7 |- ( ( F''' x ) e. B -> ( ( F''' x ) = y -> y e. B ) )
12	9, 11	syl6 34	. . . . . 6 |- ( A. x e. A ( F''' x ) e. B -> ( x e. A -> ( ( F''' x ) = y -> y e. B ) ) )
13	7, 8, 12	rexlimd 2846	. . . . 5 |- ( A. x e. A ( F''' x ) e. B -> ( E. x e. A ( F''' x ) = y -> y e. B ) )
14	6, 13	sylan9 667	. . . 4 |- ( ( F Fn A /\ A. x e. A ( F''' x ) e. B ) -> ( y e. ran F -> y e. B ) )
15	14	ssrdv 3405	. . 3 |- ( ( F Fn A /\ A. x e. A ( F''' x ) e. B ) -> ran F C_ B )
16		df-f 5564	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
17	5, 15, 16	sylanbrc 675	. 2 |- ( ( F Fn A /\ A. x e. A ( F''' x ) e. B ) -> F : A --> B )
18	4, 17	impbii 192	1 |- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F''' x ) e. B ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1447   e. wcel 1890   A.wral 2736   E.wrex 2737   C_ wss 3371   ran crn 4812   Fn wfn 5555   -->wf 5556  '''cafv 38705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-sep 4496  ax-nul 4505  ax-pr 4611

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3014  df-sbc 3235  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-if 3849  df-sn 3936  df-pr 3938  df-op 3942  df-uni 4168  df-br 4374  df-opab 4433  df-mpt 4434  df-id 4726  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-iota 5524  df-fun 5562  df-fn 5563  df-f 5564  df-fv 5568  df-dfat 38707  df-afv 38708

This theorem is referenced by:  ffnaov  38791