Theorem nfse 5089

Description: Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)

Hypotheses

Ref	Expression
nffr.r	|- F/_ x R
nffr.a	|- F/_ x A

Assertion

Ref	Expression
nfse	|- F/ x R Se A

Proof of Theorem nfse

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-se 5074	. 2 |- ( R Se A <-> A. b e. A { a e. A | a R b } e. _V )
2		nffr.a	. . 3 |- F/_ x A
3		nfcv 2764	. . . . . 6 |- F/_ x a
4		nffr.r	. . . . . 6 |- F/_ x R
5		nfcv 2764	. . . . . 6 |- F/_ x b
6	3, 4, 5	nfbr 4699	. . . . 5 |- F/ x a R b
7	6, 2	nfrab 3123	. . . 4 |- F/_ x { a e. A | a R b }
8	7	nfel1 2779	. . 3 |- F/ x { a e. A | a R b } e. _V
9	2, 8	nfral 2945	. 2 |- F/ x A. b e. A { a e. A | a R b } e. _V
10	1, 9	nfxfr 1779	1 |- F/ x R Se A

Syntax hints:   F/wnf 1708   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:  nfoi  8419