Theorem fin 5355

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

Assertion

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

Proof of Theorem fin

Step	Hyp	Ref	Expression
1		ssin 3329	. . . 4 |- ( ( ran F C_ B /\ ran F C_ C ) <-> ran F C_ ( B i^i C ) )
2	1	anbi2i 453	. . 3 |- ( ( F Fn A /\ ( ran F C_ B /\ ran F C_ C ) ) <-> ( F Fn A /\ ran F C_ ( B i^i C ) ) )
3		anandi 580	. . 3 |- ( ( 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 185	. 2 |- ( ( 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 5173	. 2 |- ( F : A --> ( B i^i C ) <-> ( F Fn A /\ ran F C_ ( B i^i C ) ) )
6		df-f 5173	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
7		df-f 5173	. . 3 |- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
8	6, 7	anbi12i 456	. 2 |- ( ( 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 211	1 |- ( F : A --> ( B i^i C ) <-> ( F : A --> B /\ F : A --> C ) )

Syntax hints:   /\ wa 103   <-> wb 104   i^i cin 3101   C_ wss 3102   ran crn 4586   Fn wfn 5164   -->wf 5165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-f 5173

This theorem is referenced by: (None)