Theorem fin 3651

Description: Mapping 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 432	. . . 4 |- ((F Fn A /\ F Fn A) <-> F Fn A)
2		ssin 2232	. . . 4 |- ((ran F (_ B /\ ran F (_ C) <-> ran F (_ (B i^i C))
3	1, 2	anbi12i 482	. . 3 |- (((F Fn A /\ F Fn A) /\ (ran F (_ B /\ ran F (_ C)) <-> (F Fn A /\ ran F (_ (B i^i C)))
4		an4 506	. . 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 175	. 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 3194	. 2 |- (F:A-->(B i^i C) <-> (F Fn A /\ ran F (_ (B i^i C)))
7		df-f 3194	. . 3 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
8		df-f 3194	. . 3 |- (F:A-->C <-> (F Fn A /\ ran F (_ C))
9	7, 8	anbi12i 482	. 2 |- ((F:A-->B /\ F:A-->C) <-> ((F Fn A /\ ran F (_ B) /\ (F Fn A /\ ran F (_ C)))
10	5, 6, 9	3bitr4 183	1 |- (F:A-->(B i^i C) <-> (F:A-->B /\ F:A-->C))

Syntax hints:   <-> wb 146   /\ wa 223   i^i cin 2046   (_ wss 2047  ran crn 3171   Fn wfn 3177  -->wf 3178

This theorem is referenced by:  lmsslem 7952

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-ss 2053  df-f 3194

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