Theorem fin 5245

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 3245	. . . 4 |- ( ( ran F C_ B /\ ran F C_ C ) <-> ran F C_ ( B i^i C ) )
2	1	anbi2i 448	. . 3 |- ( ( F Fn A /\ ( ran F C_ B /\ ran F C_ C ) ) <-> ( F Fn A /\ ran F C_ ( B i^i C ) ) )
3		anandi 560	. . 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 5063	. 2 |- ( F : A --> ( B i^i C ) <-> ( F Fn A /\ ran F C_ ( B i^i C ) ) )
6		df-f 5063	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
7		df-f 5063	. . 3 |- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
8	6, 7	anbi12i 451	. 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 3020   C_ wss 3021   ran crn 4478   Fn wfn 5054   -->wf 5055

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-in 3027  df-ss 3034  df-f 5063

This theorem is referenced by: (None)