Theorem bdinex1g 14738

Description: Bounded version of inex1g 4141. (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 3331	. . 3 |- ( x = A -> ( x i^i B ) = ( A i^i B ) )
2	1	eleq1d 2246	. 2 |- ( x = A -> ( ( x i^i B ) e. _V <-> ( A i^i B ) e. _V ) )
3		bdinex1g.bd	. . 3 |- BOUNDED B
4		vex 2742	. . 3 |- x e. _V
5	3, 4	bdinex1 14736	. 2 |- ( x i^i B ) e. _V
6	2, 5	vtoclg 2799	1 |- ( A e. V -> ( A i^i B ) e. _V )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   _Vcvv 2739   i^i cin 3130  BOUNDED wbdc 14677

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-bdsep 14721

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-v 2741  df-in 3137  df-bdc 14678

This theorem is referenced by: (None)