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Metamath Proof Explorer

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 ) )