Theorem nfse 4406

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
nfse.r	|- F/_ x R
nfse.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 4398	. 2 |- ( R Se A <-> A. b e. A { a e. A | a R b } e. _V )
2		nfse.a	. . 3 |- F/_ x A
3		nfcv 2350	. . . . . 6 |- F/_ x a
4		nfse.r	. . . . . 6 |- F/_ x R
5		nfcv 2350	. . . . . 6 |- F/_ x b
6	3, 4, 5	nfbr 4106	. . . . 5 |- F/ x a R b
7	6, 2	nfrabw 2689	. . . 4 |- F/_ x { a e. A | a R b }
8	7	nfel1 2361	. . 3 |- F/ x { a e. A | a R b } e. _V
9	2, 8	nfralxy 2546	. 2 |- F/ x A. b e. A { a e. A | a R b } e. _V
10	1, 9	nfxfr 1498	1 |- F/ x R Se A

Syntax hints:   F/wnf 1484   e. wcel 2178   F/_wnfc 2337   A.wral 2486   {crab 2490   _Vcvv 2776   class class class wbr 4059   Se wse 4394

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rab 2495  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-se 4398

This theorem is referenced by: (None)