Theorem fint 5246

Description: Function into an intersection. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 14-Oct-1999.) (Revised by set.mm contributors, 18-Sep-2011.)

Hypothesis

Ref	Expression
fint.1	|- B =/= (/)

Assertion

Ref	Expression
fint	|- (F:A-->|^|B <-> A.x e. B F:A-->x)

Distinct variable groups:   x,A   x,B   x,F

Proof of Theorem fint

Step	Hyp	Ref	Expression
1		ssint 3943	. . . 4 |- (ran F C_ |^|B <-> A.x e. B ran F C_ x)
2	1	anbi2i 675	. . 3 |- ((F Fn A /\ ran F C_ |^|B) <-> (F Fn A /\ A.x e. B ran F C_ x))
3		fint.1	. . . 4 |- B =/= (/)
4		r19.28zv 3646	. . . 4 |- (B =/= (/) -> (A.x e. B (F Fn A /\ ran F C_ x) <-> (F Fn A /\ A.x e. B ran F C_ x)))
5	3, 4	ax-mp 5	. . 3 |- (A.x e. B (F Fn A /\ ran F C_ x) <-> (F Fn A /\ A.x e. B ran F C_ x))
6	2, 5	bitr4i 243	. 2 |- ((F Fn A /\ ran F C_ |^|B) <-> A.x e. B (F Fn A /\ ran F C_ x))
7		df-f 4792	. 2 |- (F:A-->|^|B <-> (F Fn A /\ ran F C_ |^|B))
8		df-f 4792	. . 3 |- (F:A-->x <-> (F Fn A /\ ran F C_ x))
9	8	ralbii 2639	. 2 |- (A.x e. B F:A-->x <-> A.x e. B (F Fn A /\ ran F C_ x))
10	6, 7, 9	3bitr4i 268	1 |- (F:A-->|^|B <-> A.x e. B F:A-->x)

Syntax hints:   <-> wb 176   /\ wa 358   =/= wne 2517  A.wral 2615   C_ wss 3258  (/)c0 3551  |^|cint 3927  ran crn 4774   Fn wfn 4777  -->wf 4778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552  df-int 3928  df-f 4792

This theorem is referenced by: (None)