Theorem bdinex1g 10021

Description: Bounded version of inex1g 3893. (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 3131	. . 3 |- ( x = A -> ( x i^i B ) = ( A i^i B ) )
2	1	eleq1d 2106	. 2 |- ( x = A -> ( ( x i^i B ) e. _V <-> ( A i^i B ) e. _V ) )
3		bdinex1g.bd	. . 3 |- BOUNDED B
4		vex 2560	. . 3 |- x e. _V
5	3, 4	bdinex1 10019	. 2 |- ( x i^i B ) e. _V
6	2, 5	vtoclg 2613	1 |- ( A e. V -> ( A i^i B ) e. _V )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   _Vcvv 2557   i^i cin 2916  BOUNDED wbdc 9960

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-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-v 2559  df-in 2924  df-bdc 9961

This theorem is referenced by: (None)