Theorem fintm 5095

Description: Function into an intersection. (Contributed by Jim Kingdon, 28-Dec-2018.)

Hypothesis

Ref	Expression
fintm.1	|- E. x x e. B

Assertion

Ref	Expression
fintm	|- ( F : A --> |^| B <-> A. x e. B F : A --> x )

Distinct variable groups:   x, A   x, B   x, F

Proof of Theorem fintm

Step	Hyp	Ref	Expression
1		ssint 3652	. . . 4 |- ( ran F C_ |^| B <-> A. x e. B ran F C_ x )
2	1	anbi2i 444	. . 3 |- ( ( F Fn A /\ ran F C_ |^| B ) <-> ( F Fn A /\ A. x e. B ran F C_ x ) )
3		fintm.1	. . . 4 |- E. x x e. B
4		r19.28mv 3334	. . . 4 |- ( E. x x e. B -> ( A. x e. B ( F Fn A /\ ran F C_ x ) <-> ( F Fn A /\ A. x e. B ran F C_ x ) ) )
5	3, 4	ax-mp 7	. . 3 |- ( A. x e. B ( F Fn A /\ ran F C_ x ) <-> ( F Fn A /\ A. x e. B ran F C_ x ) )
6	2, 5	bitr4i 185	. 2 |- ( ( F Fn A /\ ran F C_ |^| B ) <-> A. x e. B ( F Fn A /\ ran F C_ x ) )
7		df-f 4926	. 2 |- ( F : A --> |^| B <-> ( F Fn A /\ ran F C_ |^| B ) )
8		df-f 4926	. . 3 |- ( F : A --> x <-> ( F Fn A /\ ran F C_ x ) )
9	8	ralbii 2372	. 2 |- ( A. x e. B F : A --> x <-> A. x e. B ( F Fn A /\ ran F C_ x ) )
10	6, 7, 9	3bitr4i 210	1 |- ( F : A --> |^| B <-> A. x e. B F : A --> x )

Syntax hints:   /\ wa 102   <-> wb 103   E.wex 1421   e. wcel 1433   A.wral 2348   C_ wss 2973   |^|cint 3636   ran crn 4364   Fn wfn 4917   -->wf 4918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-in 2979  df-ss 2986  df-int 3637  df-f 4926

This theorem is referenced by: (None)