Theorem fintm 5511

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 3939	. . . 4 |- ( ran F C_ |^| B <-> A. x e. B ran F C_ x )
2	1	anbi2i 457	. . 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 3584	. . . 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 187	. 2 |- ( ( F Fn A /\ ran F C_ |^| B ) <-> A. x e. B ( F Fn A /\ ran F C_ x ) )
7		df-f 5322	. 2 |- ( F : A --> |^| B <-> ( F Fn A /\ ran F C_ |^| B ) )
8		df-f 5322	. . 3 |- ( F : A --> x <-> ( F Fn A /\ ran F C_ x ) )
9	8	ralbii 2536	. 2 |- ( A. x e. B F : A --> x <-> A. x e. B ( F Fn A /\ ran F C_ x ) )
10	6, 7, 9	3bitr4i 212	1 |- ( F : A --> |^| B <-> A. x e. B F : A --> x )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1538   e. wcel 2200   A.wral 2508   C_ wss 3197   |^|cint 3923   ran crn 4720   Fn wfn 5313   -->wf 5314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-in 3203  df-ss 3210  df-int 3924  df-f 5322

This theorem is referenced by: (None)