Theorem bj-inex 10027

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 2568	. 2 |- ( A e. V -> E. x x = A )
2		elisset 2568	. 2 |- ( B e. W -> E. y y = B )
3		ax-17 1419	. . . 4 |- ( E. y y = B -> A. x E. y y = B )
4		19.29r 1512	. . . 4 |- ( ( E. x x = A /\ A. x E. y y = B ) -> E. x ( x = A /\ E. y y = B ) )
5	3, 4	sylan2 270	. . 3 |- ( ( E. x x = A /\ E. y y = B ) -> E. x ( x = A /\ E. y y = B ) )
6		ax-17 1419	. . . . 5 |- ( x = A -> A. y x = A )
7		19.29 1511	. . . . 5 |- ( ( A. y x = A /\ E. y y = B ) -> E. y ( x = A /\ y = B ) )
8	6, 7	sylan 267	. . . 4 |- ( ( x = A /\ E. y y = B ) -> E. y ( x = A /\ y = B ) )
9	8	eximi 1491	. . 3 |- ( E. x ( x = A /\ E. y y = B ) -> E. x E. y ( x = A /\ y = B ) )
10		ineq12 3133	. . . . 5 |- ( ( x = A /\ y = B ) -> ( x i^i y ) = ( A i^i B ) )
11	10	2eximi 1492	. . . 4 |- ( E. x E. y ( x = A /\ y = B ) -> E. x E. y ( x i^i y ) = ( A i^i B ) )
12		dfin5 2925	. . . . . . 7 |- ( x i^i y ) = { z e. x | z e. y }
13		vex 2560	. . . . . . . 8 |- x e. _V
14		ax-bdel 9941	. . . . . . . . 9 |- BOUNDED z e. y
15		bdcv 9968	. . . . . . . . 9 |- BOUNDED x
16	14, 15	bdrabexg 10026	. . . . . . . 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 2110	. . . . . 6 |- ( x i^i y ) e. _V
19		eleq1 2100	. . . . . 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 136	. . . . 5 |- ( ( x i^i y ) = ( A i^i B ) -> ( A i^i B ) e. _V )
21	20	exlimivv 1776	. . . 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 273	1 |- ( ( A e. V /\ B e. W ) -> ( A i^i B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   {crab 2310   _Vcvv 2557   i^i cin 2916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bd0 9933  ax-bdan 9935  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-bdc 9961

This theorem is referenced by:  speano5  10069