Theorem bj-inex 10698

Description: The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-inex	|- ( ( A e. V /\ B e. W ) -> ( A i^i B ) e. _V )

Proof of Theorem bj-inex

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elisset 2613	. 2 |- ( A e. V -> E. x x = A )
2		elisset 2613	. 2 |- ( B e. W -> E. y y = B )
3		ax-17 1459	. . . 4 |- ( E. y y = B -> A. x E. y y = B )
4		19.29r 1552	. . . 4 |- ( ( E. x x = A /\ A. x E. y y = B ) -> E. x ( x = A /\ E. y y = B ) )
5	3, 4	sylan2 280	. . 3 |- ( ( E. x x = A /\ E. y y = B ) -> E. x ( x = A /\ E. y y = B ) )
6		ax-17 1459	. . . . 5 |- ( x = A -> A. y x = A )
7		19.29 1551	. . . . 5 |- ( ( A. y x = A /\ E. y y = B ) -> E. y ( x = A /\ y = B ) )
8	6, 7	sylan 277	. . . 4 |- ( ( x = A /\ E. y y = B ) -> E. y ( x = A /\ y = B ) )
9	8	eximi 1531	. . 3 |- ( E. x ( x = A /\ E. y y = B ) -> E. x E. y ( x = A /\ y = B ) )
10		ineq12 3162	. . . . 5 |- ( ( x = A /\ y = B ) -> ( x i^i y ) = ( A i^i B ) )
11	10	2eximi 1532	. . . 4 |- ( E. x E. y ( x = A /\ y = B ) -> E. x E. y ( x i^i y ) = ( A i^i B ) )
12		dfin5 2980	. . . . . . 7 |- ( x i^i y ) = { z e. x | z e. y }
13		vex 2604	. . . . . . . 8 |- x e. _V
14		ax-bdel 10612	. . . . . . . . 9 |- BOUNDED z e. y
15		bdcv 10639	. . . . . . . . 9 |- BOUNDED x
16	14, 15	bdrabexg 10697	. . . . . . . 8 |- ( x e. _V -> { z e. x | z e. y } e. _V )
17	13, 16	ax-mp 7	. . . . . . 7 |- { z e. x | z e. y } e. _V
18	12, 17	eqeltri 2151	. . . . . 6 |- ( x i^i y ) e. _V
19		eleq1 2141	. . . . . 6 |- ( ( x i^i y ) = ( A i^i B ) -> ( ( x i^i y ) e. _V <-> ( A i^i B ) e. _V ) )
20	18, 19	mpbii 146	. . . . 5 |- ( ( x i^i y ) = ( A i^i B ) -> ( A i^i B ) e. _V )
21	20	exlimivv 1817	. . . 4 |- ( E. x E. y ( x i^i y ) = ( A i^i B ) -> ( A i^i B ) e. _V )
22	11, 21	syl 14	. . 3 |- ( E. x E. y ( x = A /\ y = B ) -> ( A i^i B ) e. _V )
23	5, 9, 22	3syl 17	. 2 |- ( ( E. x x = A /\ E. y y = B ) -> ( A i^i B ) e. _V )
24	1, 2, 23	syl2an 283	1 |- ( ( A e. V /\ B e. W ) -> ( A i^i B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   {crab 2352   _Vcvv 2601   i^i cin 2972

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-bd0 10604  ax-bdan 10606  ax-bdel 10612  ax-bdsb 10613  ax-bdsep 10675

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603  df-in 2979  df-ss 2986  df-bdc 10632

This theorem is referenced by:  speano5  10739