Theorem nfse 4192

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 4184	. 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 2235	. . . . . 6 |- F/_ x a
4		nfse.r	. . . . . 6 |- F/_ x R
5		nfcv 2235	. . . . . 6 |- F/_ x b
6	3, 4, 5	nfbr 3911	. . . . 5 |- F/ x a R b
7	6, 2	nfrabxy 2561	. . . 4 |- F/_ x { a e. A | a R b }
8	7	nfel1 2246	. . 3 |- F/ x { a e. A | a R b } e. _V
9	2, 8	nfralxy 2425	. 2 |- F/ x A. b e. A { a e. A | a R b } e. _V
10	1, 9	nfxfr 1415	1 |- F/ x R Se A

Syntax hints:   F/wnf 1401   e. wcel 1445   F/_wnfc 2222   A.wral 2370   {crab 2374   _Vcvv 2633   class class class wbr 3867   Se wse 4180

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rab 2379  df-v 2635  df-un 3017  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-se 4184

This theorem is referenced by: (None)