Theorem fint 3656

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 2352	. . . . 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 2553	. . . 4 |- (ran F (_ |^|B <-> A.x e. B ran F (_ x)
5	3, 4	anbi12i 484	. . 3 |- ((F Fn A /\ ran F (_ |^|B) <-> (A.x e. B F Fn A /\ A.x e. B ran F (_ x))
6		r19.26 1753	. . 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 3200	. 2 |- (F:A-->|^|B <-> (F Fn A /\ ran F (_ |^|B))
9		df-f 3200	. . 3 |- (F:A-->x <-> (F Fn A /\ ran F (_ x))
10	9	ralbii 1670	. 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 1588  A.wral 1648   (_ wss 2050  (/)c0 2283  |^|cint 2537  ran crn 3177   Fn wfn 3183  -->wf 3184

This theorem is referenced by:  chintcl 9290

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-in 2054  df-ss 2056  df-nul 2284  df-int 2538  df-f 3200

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