Theorem bj-inex 14698

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 2753	. 2 |- ( A e. V -> E. x x = A )
2		elisset 2753	. 2 |- ( B e. W -> E. y y = B )
3		ax-17 1526	. . . 4 |- ( E. y y = B -> A. x E. y y = B )
4		19.29r 1621	. . . 4 |- ( ( E. x x = A /\ A. x E. y y = B ) -> E. x ( x = A /\ E. y y = B ) )
5	3, 4	sylan2 286	. . 3 |- ( ( E. x x = A /\ E. y y = B ) -> E. x ( x = A /\ E. y y = B ) )
6		ax-17 1526	. . . . 5 |- ( x = A -> A. y x = A )
7		19.29 1620	. . . . 5 |- ( ( A. y x = A /\ E. y y = B ) -> E. y ( x = A /\ y = B ) )
8	6, 7	sylan 283	. . . 4 |- ( ( x = A /\ E. y y = B ) -> E. y ( x = A /\ y = B ) )
9	8	eximi 1600	. . 3 |- ( E. x ( x = A /\ E. y y = B ) -> E. x E. y ( x = A /\ y = B ) )
10		ineq12 3333	. . . . 5 |- ( ( x = A /\ y = B ) -> ( x i^i y ) = ( A i^i B ) )
11	10	2eximi 1601	. . . 4 |- ( E. x E. y ( x = A /\ y = B ) -> E. x E. y ( x i^i y ) = ( A i^i B ) )
12		dfin5 3138	. . . . . . 7 |- ( x i^i y ) = { z e. x | z e. y }
13		vex 2742	. . . . . . . 8 |- x e. _V
14		ax-bdel 14612	. . . . . . . . 9 |- BOUNDED z e. y
15		bdcv 14639	. . . . . . . . 9 |- BOUNDED x
16	14, 15	bdrabexg 14697	. . . . . . . 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 2250	. . . . . 6 |- ( x i^i y ) e. _V
19		eleq1 2240	. . . . . 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 148	. . . . 5 |- ( ( x i^i y ) = ( A i^i B ) -> ( A i^i B ) e. _V )
21	20	exlimivv 1896	. . . 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 289	1 |- ( ( A e. V /\ B e. W ) -> ( A i^i B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1351   = wceq 1353   E.wex 1492   e. wcel 2148   {crab 2459   _Vcvv 2739   i^i cin 3130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bd0 14604  ax-bdan 14606  ax-bdel 14612  ax-bdsb 14613  ax-bdsep 14675

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2741  df-in 3137  df-ss 3144  df-bdc 14632

This theorem is referenced by:  speano5  14735