Theorem fin 5531

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 3431	. . . 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 594	. . 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 5337	. 2 |- ( F : A --> ( B i^i C ) <-> ( F Fn A /\ ran F C_ ( B i^i C ) ) )
6		df-f 5337	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
7		df-f 5337	. . 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 3200   C_ wss 3201   ran crn 4732   Fn wfn 5328   -->wf 5329

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-ss 3214  df-f 5337

This theorem is referenced by:  umgrislfupgrdom  16072  usgrislfuspgrdom  16131