Theorem fintm 5075

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 3631	. . . 4 |- ( ran F C_ |^| B <-> A. x e. B ran F C_ x )
2	1	anbi2i 430	. . 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 3314	. . . 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 176	. 2 |- ( ( F Fn A /\ ran F C_ |^| B ) <-> A. x e. B ( F Fn A /\ ran F C_ x ) )
7		df-f 4906	. 2 |- ( F : A --> |^| B <-> ( F Fn A /\ ran F C_ |^| B ) )
8		df-f 4906	. . 3 |- ( F : A --> x <-> ( F Fn A /\ ran F C_ x ) )
9	8	ralbii 2330	. 2 |- ( A. x e. B F : A --> x <-> A. x e. B ( F Fn A /\ ran F C_ x ) )
10	6, 7, 9	3bitr4i 201	1 |- ( F : A --> |^| B <-> A. x e. B F : A --> x )

Syntax hints:   /\ wa 97   <-> wb 98   E.wex 1381   e. wcel 1393   A.wral 2306   C_ wss 2917   |^|cint 3615   ran crn 4346   Fn wfn 4897   -->wf 4898

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-int 3616  df-f 4906

This theorem is referenced by: (None)