Theorem fin

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

	Ref	Expression
Assertion	fin	|- ( F : A --> ( B i^i C ) <-> ( F : A --> B /\ F : A --> C ) )

Proof

Step	Hyp	Ref	Expression
1		ssin	|- ( ( ran F C_ B /\ ran F C_ C ) <-> ran F C_ ( B i^i C ) )
2	1	anbi2i	|- ( ( F Fn A /\ ( ran F C_ B /\ ran F C_ C ) ) <-> ( F Fn A /\ ran F C_ ( B i^i C ) ) )
3		anandi	|- ( ( F Fn A /\ ( ran F C_ B /\ ran F C_ C ) ) <-> ( ( F Fn A /\ ran F C_ B ) /\ ( F Fn A /\ ran F C_ C ) ) )
4	2 3	bitr3i	|- ( ( F Fn A /\ ran F C_ ( B i^i C ) ) <-> ( ( F Fn A /\ ran F C_ B ) /\ ( F Fn A /\ ran F C_ C ) ) )
5		df-f	|- ( F : A --> ( B i^i C ) <-> ( F Fn A /\ ran F C_ ( B i^i C ) ) )
6		df-f	|- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
7		df-f	|- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
8	6 7	anbi12i	|- ( ( F : A --> B /\ F : A --> C ) <-> ( ( F Fn A /\ ran F C_ B ) /\ ( F Fn A /\ ran F C_ C ) ) )
9	4 5 8	3bitr4i	|- ( F : A --> ( B i^i C ) <-> ( F : A --> B /\ F : A --> C ) )