Theorem fin 5417

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 3372	. . . 4 |- ( ( ran F C_ B /\ ran F C_ C ) <-> ran F C_ ( B i^i C ) )
2	1	anbi2i 457	. . 3 |- ( ( F Fn A /\ ( ran F C_ B /\ ran F C_ C ) ) <-> ( F Fn A /\ ran F C_ ( B i^i C ) ) )
3		anandi 590	. . 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 186	. 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 5235	. 2 |- ( F : A --> ( B i^i C ) <-> ( F Fn A /\ ran F C_ ( B i^i C ) ) )
6		df-f 5235	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
7		df-f 5235	. . 3 |- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
8	6, 7	anbi12i 460	. 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 212	1 |- ( F : A --> ( B i^i C ) <-> ( F : A --> B /\ F : A --> C ) )

Syntax hints:   /\ wa 104   <-> wb 105   i^i cin 3143   C_ wss 3144   ran crn 4642   Fn wfn 5226   -->wf 5227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-f 5235

This theorem is referenced by: (None)