Theorem nfse 4319

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 4311	. 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 2308	. . . . . 6 |- F/_ x a
4		nfse.r	. . . . . 6 |- F/_ x R
5		nfcv 2308	. . . . . 6 |- F/_ x b
6	3, 4, 5	nfbr 4028	. . . . 5 |- F/ x a R b
7	6, 2	nfrabxy 2646	. . . 4 |- F/_ x { a e. A | a R b }
8	7	nfel1 2319	. . 3 |- F/ x { a e. A | a R b } e. _V
9	2, 8	nfralxy 2504	. 2 |- F/ x A. b e. A { a e. A | a R b } e. _V
10	1, 9	nfxfr 1462	1 |- F/ x R Se A

Syntax hints:   F/wnf 1448   e. wcel 2136   F/_wnfc 2295   A.wral 2444   {crab 2448   _Vcvv 2726   class class class wbr 3982   Se wse 4307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-se 4311

This theorem is referenced by: (None)