Theorem nfse 4168

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	⊢ Ⅎ𝑥𝑅
nfse.a	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfse	⊢ Ⅎ𝑥 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 4160	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑏 ∈ 𝐴 {𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V)
2		nfse.a	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcv 2228	. . . . . 6 ⊢ Ⅎ𝑥𝑎
4		nfse.r	. . . . . 6 ⊢ Ⅎ𝑥𝑅
5		nfcv 2228	. . . . . 6 ⊢ Ⅎ𝑥𝑏
6	3, 4, 5	nfbr 3889	. . . . 5 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
7	6, 2	nfrabxy 2547	. . . 4 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏}
8	7	nfel1 2239	. . 3 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V
9	2, 8	nfralxy 2414	. 2 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 {𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V
10	1, 9	nfxfr 1408	1 ⊢ Ⅎ𝑥 𝑅 Se 𝐴

Syntax hints:  Ⅎwnf 1394   ∈ wcel 1438  Ⅎwnfc 2215  ∀wral 2359  {crab 2363  Vcvv 2619   class class class wbr 3845   Se wse 4156

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-se 4160

This theorem is referenced by: (None)