Theorem dfafn5b 41241

Description: Representation of a function in terms of its values, analogous to dffn5 6241 (only if it is assumed that the function value for each x is a set). (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
dfafn5b	|- ( A. x e. A ( F''' x ) e. V -> ( F Fn A <-> F = ( x e. A |-> ( F''' x ) ) ) )

Distinct variable groups:   x, A   x, F

Allowed substitution hint:   V( x)

Proof of Theorem dfafn5b

Step	Hyp	Ref	Expression
1		dfafn5a 41240	. 2 |- ( F Fn A -> F = ( x e. A |-> ( F''' x ) ) )
2		eqid 2622	. . . 4 |- ( x e. A |-> ( F''' x ) ) = ( x e. A |-> ( F''' x ) )
3	2	fnmpt 6020	. . 3 |- ( A. x e. A ( F''' x ) e. V -> ( x e. A |-> ( F''' x ) ) Fn A )
4		fneq1 5979	. . 3 |- ( F = ( x e. A |-> ( F''' x ) ) -> ( F Fn A <-> ( x e. A |-> ( F''' x ) ) Fn A ) )
5	3, 4	syl5ibrcom 237	. 2 |- ( A. x e. A ( F''' x ) e. V -> ( F = ( x e. A |-> ( F''' x ) ) -> F Fn A ) )
6	1, 5	impbid2 216	1 |- ( A. x e. A ( F''' x ) e. V -> ( F Fn A <-> F = ( x e. A |-> ( F''' x ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   |-> cmpt 4729   Fn wfn 5883  '''cafv 41194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-dfat 41196  df-afv 41197

This theorem is referenced by: (None)