Theorem bj-inex 13789

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 2740	. 2 |- ( A e. V -> E. x x = A )
2		elisset 2740	. 2 |- ( B e. W -> E. y y = B )
3		ax-17 1514	. . . 4 |- ( E. y y = B -> A. x E. y y = B )
4		19.29r 1609	. . . 4 |- ( ( E. x x = A /\ A. x E. y y = B ) -> E. x ( x = A /\ E. y y = B ) )
5	3, 4	sylan2 284	. . 3 |- ( ( E. x x = A /\ E. y y = B ) -> E. x ( x = A /\ E. y y = B ) )
6		ax-17 1514	. . . . 5 |- ( x = A -> A. y x = A )
7		19.29 1608	. . . . 5 |- ( ( A. y x = A /\ E. y y = B ) -> E. y ( x = A /\ y = B ) )
8	6, 7	sylan 281	. . . 4 |- ( ( x = A /\ E. y y = B ) -> E. y ( x = A /\ y = B ) )
9	8	eximi 1588	. . 3 |- ( E. x ( x = A /\ E. y y = B ) -> E. x E. y ( x = A /\ y = B ) )
10		ineq12 3318	. . . . 5 |- ( ( x = A /\ y = B ) -> ( x i^i y ) = ( A i^i B ) )
11	10	2eximi 1589	. . . 4 |- ( E. x E. y ( x = A /\ y = B ) -> E. x E. y ( x i^i y ) = ( A i^i B ) )
12		dfin5 3123	. . . . . . 7 |- ( x i^i y ) = { z e. x | z e. y }
13		vex 2729	. . . . . . . 8 |- x e. _V
14		ax-bdel 13703	. . . . . . . . 9 |- BOUNDED z e. y
15		bdcv 13730	. . . . . . . . 9 |- BOUNDED x
16	14, 15	bdrabexg 13788	. . . . . . . 8 |- ( x e. _V -> { z e. x | z e. y } e. _V )
17	13, 16	ax-mp 5	. . . . . . 7 |- { z e. x | z e. y } e. _V
18	12, 17	eqeltri 2239	. . . . . 6 |- ( x i^i y ) e. _V
19		eleq1 2229	. . . . . 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 147	. . . . 5 |- ( ( x i^i y ) = ( A i^i B ) -> ( A i^i B ) e. _V )
21	20	exlimivv 1884	. . . 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 287	1 |- ( ( A e. V /\ B e. W ) -> ( A i^i B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1341   = wceq 1343   E.wex 1480   e. wcel 2136   {crab 2448   _Vcvv 2726   i^i cin 3115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-bd0 13695  ax-bdan 13697  ax-bdel 13703  ax-bdsb 13704  ax-bdsep 13766

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453  df-v 2728  df-in 3122  df-ss 3129  df-bdc 13723

This theorem is referenced by:  speano5  13826