Theorem nfse 4263

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 4255	. 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 2281	. . . . . 6 |- F/_ x a
4		nfse.r	. . . . . 6 |- F/_ x R
5		nfcv 2281	. . . . . 6 |- F/_ x b
6	3, 4, 5	nfbr 3974	. . . . 5 |- F/ x a R b
7	6, 2	nfrabxy 2611	. . . 4 |- F/_ x { a e. A | a R b }
8	7	nfel1 2292	. . 3 |- F/ x { a e. A | a R b } e. _V
9	2, 8	nfralxy 2471	. 2 |- F/ x A. b e. A { a e. A | a R b } e. _V
10	1, 9	nfxfr 1450	1 |- F/ x R Se A

Syntax hints:   F/wnf 1436   e. wcel 1480   F/_wnfc 2268   A.wral 2416   {crab 2420   _Vcvv 2686   class class class wbr 3929   Se wse 4251

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-se 4255

This theorem is referenced by: (None)