Theorem fint 6084

Description: Function into an intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Hypothesis

Ref	Expression
fint.1	|- B =/= (/)

Assertion

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

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

Proof of Theorem fint

Step	Hyp	Ref	Expression
1		ssint 4493	. . . 4 |- ( ran F C_ |^| B <-> A. x e. B ran F C_ x )
2	1	anbi2i 730	. . 3 |- ( ( F Fn A /\ ran F C_ |^| B ) <-> ( F Fn A /\ A. x e. B ran F C_ x ) )
3		fint.1	. . . 4 |- B =/= (/)
4		r19.28zv 4066	. . . 4 |- ( 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 267	. 2 |- ( ( F Fn A /\ ran F C_ |^| B ) <-> A. x e. B ( F Fn A /\ ran F C_ x ) )
7		df-f 5892	. 2 |- ( F : A --> |^| B <-> ( F Fn A /\ ran F C_ |^| B ) )
8		df-f 5892	. . 3 |- ( F : A --> x <-> ( F Fn A /\ ran F C_ x ) )
9	8	ralbii 2980	. 2 |- ( A. x e. B F : A --> x <-> A. x e. B ( F Fn A /\ ran F C_ x ) )
10	6, 7, 9	3bitr4i 292	1 |- ( F : A --> |^| B <-> A. x e. B F : A --> x )

Syntax hints:   <-> wb 196   /\ wa 384   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   |^|cint 4475   ran crn 5115   Fn wfn 5883   -->wf 5884

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-int 4476  df-f 5892

This theorem is referenced by:  chintcli  28190