Theorem bj-seex 32919

Description: Version of seex 5077 with a dv condition replaced by a non-freeness hypothesis (for the sake of illustration). (Contributed by BJ, 27-Apr-2019.)

Hypothesis

Ref	Expression
bj-seex.nf	|- F/_ x B

Assertion

Ref	Expression
bj-seex	|- ( ( R Se A /\ B e. A ) -> { x e. A | x R B } e. _V )

Distinct variable groups:   x, A   x, R

Allowed substitution hint:   B( x)

Proof of Theorem bj-seex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-se 5074	. 2 |- ( R Se A <-> A. y e. A { x e. A | x R y } e. _V )
2		bj-seex.nf	. . . . . 6 |- F/_ x B
3	2	nfeq2 2780	. . . . 5 |- F/ x y = B
4		breq2 4657	. . . . 5 |- ( y = B -> ( x R y <-> x R B ) )
5	3, 4	bj-rabbid 32915	. . . 4 |- ( y = B -> { x e. A | x R y } = { x e. A | x R B } )
6	5	eleq1d 2686	. . 3 |- ( y = B -> ( { x e. A | x R y } e. _V <-> { x e. A | x R B } e. _V ) )
7	6	rspccva 3308	. 2 |- ( ( A. y e. A { x e. A | x R y } e. _V /\ B e. A ) -> { x e. A | x R B } e. _V )
8	1, 7	sylanb 489	1 |- ( ( R Se A /\ B e. A ) -> { x e. A | x R B } e. _V )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   A.wral 2912   {crab 2916   _Vcvv 3200   class class class wbr 4653   Se wse 5071

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  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-br 4654  df-se 5074

This theorem is referenced by: (None)