Theorem fintm 5303

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 3782	. . . 4 |- ( ran F C_ |^| B <-> A. x e. B ran F C_ x )
2	1	anbi2i 452	. . 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 3450	. . . 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 5	. . 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 186	. 2 |- ( ( F Fn A /\ ran F C_ |^| B ) <-> A. x e. B ( F Fn A /\ ran F C_ x ) )
7		df-f 5122	. 2 |- ( F : A --> |^| B <-> ( F Fn A /\ ran F C_ |^| B ) )
8		df-f 5122	. . 3 |- ( F : A --> x <-> ( F Fn A /\ ran F C_ x ) )
9	8	ralbii 2439	. 2 |- ( A. x e. B F : A --> x <-> A. x e. B ( F Fn A /\ ran F C_ x ) )
10	6, 7, 9	3bitr4i 211	1 |- ( F : A --> |^| B <-> A. x e. B F : A --> x )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1468   e. wcel 1480   A.wral 2414   C_ wss 3066   |^|cint 3766   ran crn 4535   Fn wfn 5113   -->wf 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-in 3072  df-ss 3079  df-int 3767  df-f 5122

This theorem is referenced by: (None)