Theorem nfse 4271

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 4263	. 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 2282	. . . . . 6 |- F/_ x a
4		nfse.r	. . . . . 6 |- F/_ x R
5		nfcv 2282	. . . . . 6 |- F/_ x b
6	3, 4, 5	nfbr 3982	. . . . 5 |- F/ x a R b
7	6, 2	nfrabxy 2614	. . . 4 |- F/_ x { a e. A | a R b }
8	7	nfel1 2293	. . 3 |- F/ x { a e. A | a R b } e. _V
9	2, 8	nfralxy 2474	. 2 |- F/ x A. b e. A { a e. A | a R b } e. _V
10	1, 9	nfxfr 1451	1 |- F/ x R Se A

Syntax hints:   F/wnf 1437   e. wcel 1481   F/_wnfc 2269   A.wral 2417   {crab 2421   _Vcvv 2689   class class class wbr 3937   Se wse 4259

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-se 4263

This theorem is referenced by: (None)