Theorem bj-clex 32952

Description: Sethood of certain classes. (Contributed by BJ, 2-Apr-2019.)

Assertion

Ref	Expression
bj-clex	|- ( A e. V -> { x | { x } e. ( A " B ) } e. _V )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   V( x)

Proof of Theorem bj-clex

Step	Hyp	Ref	Expression
1		imaexg 7103	. 2 |- ( A e. V -> ( A " B ) e. _V )
2		bj-snsetex 32951	. 2 |- ( ( A " B ) e. _V -> { x | { x } e. ( A " B ) } e. _V )
3	1, 2	syl 17	1 |- ( A e. V -> { x | { x } e. ( A " B ) } e. _V )

Syntax hints:   -> wi 4   e. wcel 1990   {cab 2608   _Vcvv 3200   {csn 4177   "cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  bj-projex  32983