Theorem ffnafv 32495

Description: A function maps to a class to which all values belong, analogous to ffnfv 6033. (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 5713	. . 3 |- ( F : A --> B -> F Fn A )
2		fafvelrn 32494	. . . 4 |- ( ( F : A --> B /\ x e. A ) -> ( F''' x ) e. B )
3	2	ralrimiva 2868	. . 3 |- ( F : A --> B -> A. x e. A ( F''' x ) e. B )
4	1, 3	jca 530	. 2 |- ( F : A --> B -> ( F Fn A /\ A. x e. A ( F''' x ) e. B ) )
5		simpl 455	. . 3 |- ( ( F Fn A /\ A. x e. A ( F''' x ) e. B ) -> F Fn A )
6		afvelrnb0 32488	. . . . 5 |- ( F Fn A -> ( y e. ran F -> E. x e. A ( F''' x ) = y ) )
7		nfra1 2835	. . . . . 6 |- F/ x A. x e. A ( F''' x ) e. B
8		nfv 1712	. . . . . 6 |- F/ x y e. B
9		rsp 2820	. . . . . . 7 |- ( A. x e. A ( F''' x ) e. B -> ( x e. A -> ( F''' x ) e. B ) )
10		eleq1 2526	. . . . . . . 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 2938	. . . . 5 |- ( A. x e. A ( F''' x ) e. B -> ( E. x e. A ( F''' x ) = y -> y e. B ) )
14	6, 13	sylan9 655	. . . 4 |- ( ( F Fn A /\ A. x e. A ( F''' x ) e. B ) -> ( y e. ran F -> y e. B ) )
15	14	ssrdv 3495	. . 3 |- ( ( F Fn A /\ A. x e. A ( F''' x ) e. B ) -> ran F C_ B )
16		df-f 5574	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
17	5, 15, 16	sylanbrc 662	. 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 367   = wceq 1398   e. wcel 1823   A.wral 2804   E.wrex 2805   C_ wss 3461   ran crn 4989   Fn wfn 5565   -->wf 5566  '''cafv 32438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-dfat 32440  df-afv 32441

This theorem is referenced by:  ffnaov  32523