Theorem bdinex1g 13783

Description: Bounded version of inex1g 4118. (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 3316	. . 3 |- ( x = A -> ( x i^i B ) = ( A i^i B ) )
2	1	eleq1d 2235	. 2 |- ( x = A -> ( ( x i^i B ) e. _V <-> ( A i^i B ) e. _V ) )
3		bdinex1g.bd	. . 3 |- BOUNDED B
4		vex 2729	. . 3 |- x e. _V
5	3, 4	bdinex1 13781	. 2 |- ( x i^i B ) e. _V
6	2, 5	vtoclg 2786	1 |- ( A e. V -> ( A i^i B ) e. _V )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   _Vcvv 2726   i^i cin 3115  BOUNDED wbdc 13722

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-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-v 2728  df-in 3122  df-bdc 13723

This theorem is referenced by: (None)