Theorem fin 2770

Description: Function into an intersection.

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		anidm 331	. . . 4 |- ((F Fn A /\ F Fn A) <-> F Fn A)
2		ssin 1659	. . . 4 |- ((ran F (_ B /\ ran F (_ C) <-> ran F (_ (B i^i C))
3	1, 2	anbi12i 369	. . 3 |- (((F Fn A /\ F Fn A) /\ (ran F (_ B /\ ran F (_ C)) <-> (F Fn A /\ ran F (_ (B i^i C)))
4		an4 388	. . 3 |- (((F Fn A /\ F Fn A) /\ (ran F (_ B /\ ran F (_ C)) <-> ((F Fn A /\ ran F (_ B) /\ (F Fn A /\ ran F (_ C)))
5	3, 4	bitr3 153	. 2 |- ((F Fn A /\ ran F (_ (B i^i C)) <-> ((F Fn A /\ ran F (_ B) /\ (F Fn A /\ ran F (_ C)))
6		df-f 2434	. 2 |- (F:A-->(B i^i C) <-> (F Fn A /\ ran F (_ (B i^i C)))
7		df-f 2434	. . 3 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
8		df-f 2434	. . 3 |- (F:A-->C <-> (F Fn A /\ ran F (_ C))
9	7, 8	anbi12i 369	. 2 |- ((F:A-->B /\ F:A-->C) <-> ((F Fn A /\ ran F (_ B) /\ (F Fn A /\ ran F (_ C)))
10	5, 6, 9	3bitr4 158	1 |- (F:A-->(B i^i C) <-> (F:A-->B /\ F:A-->C))

Syntax hints:   <-> wb 127   /\ wa 196   i^i cin 1486   (_ wss 1487  ran crn 2411   Fn wfn 2417  -->wf 2418

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492  df-f 2434

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