Theorem fin 5512

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 3426	. . . 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 592	. . 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 5322	. 2 |- ( F : A --> ( B i^i C ) <-> ( F Fn A /\ ran F C_ ( B i^i C ) ) )
6		df-f 5322	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
7		df-f 5322	. . 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 3196   C_ wss 3197   ran crn 4720   Fn wfn 5313   -->wf 5314

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-f 5322

This theorem is referenced by:  umgrislfupgrdom  15929  usgrislfuspgrdom  15988