Theorem bdinex1g 16264

Description: Bounded version of inex1g 4220. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bdinex1g.bd	|- BOUNDED B

Assertion

Ref	Expression
bdinex1g	|- ( A e. V -> ( A i^i B ) e. _V )

Proof of Theorem bdinex1g

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ineq1 3398	. . 3 |- ( x = A -> ( x i^i B ) = ( A i^i B ) )
2	1	eleq1d 2298	. 2 |- ( x = A -> ( ( x i^i B ) e. _V <-> ( A i^i B ) e. _V ) )
3		bdinex1g.bd	. . 3 |- BOUNDED B
4		vex 2802	. . 3 |- x e. _V
5	3, 4	bdinex1 16262	. 2 |- ( x i^i B ) e. _V
6	2, 5	vtoclg 2861	1 |- ( A e. V -> ( A i^i B ) e. _V )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   i^i cin 3196  BOUNDED wbdc 16203

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-bdsep 16247

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-bdc 16204

This theorem is referenced by: (None)