Theorem fint 3650

Description: Function into an intersection.

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		fint.1	. . . . 5 |- B =/= (/)
2		r19.3rzv 2348	. . . . 5 |- (B =/= (/) -> (F Fn A <-> A.x e. B F Fn A))
3	1, 2	ax-mp 7	. . . 4 |- (F Fn A <-> A.x e. B F Fn A)
4		ssint 2549	. . . 4 |- (ran F (_ |^|B <-> A.x e. B ran F (_ x)
5	3, 4	anbi12i 482	. . 3 |- ((F Fn A /\ ran F (_ |^|B) <-> (A.x e. B F Fn A /\ A.x e. B ran F (_ x))
6		r19.26 1750	. . 3 |- (A.x e. B (F Fn A /\ ran F (_ x) <-> (A.x e. B F Fn A /\ A.x e. B ran F (_ x))
7	5, 6	bitr4 176	. 2 |- ((F Fn A /\ ran F (_ |^|B) <-> A.x e. B (F Fn A /\ ran F (_ x))
8		df-f 3194	. 2 |- (F:A-->|^|B <-> (F Fn A /\ ran F (_ |^|B))
9		df-f 3194	. . 3 |- (F:A-->x <-> (F Fn A /\ ran F (_ x))
10	9	ralbii 1667	. 2 |- (A.x e. B F:A-->x <-> A.x e. B (F Fn A /\ ran F (_ x))
11	7, 8, 10	3bitr4 183	1 |- (F:A-->|^|B <-> A.x e. B F:A-->x)

Syntax hints:   <-> wb 146   /\ wa 223   =/= wne 1585  A.wral 1645   (_ wss 2047  (/)c0 2280  |^|cint 2533  ran crn 3171   Fn wfn 3177  -->wf 3178

This theorem is referenced by:  chintcl 9295

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-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-nul 2281  df-int 2534  df-f 3194

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