Theorem nfse 5564

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
nffr.r	⊢ Ⅎ𝑥𝑅
nffr.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 5545	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑏 ∈ 𝐴 {𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V)
2		nffr.a	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑥𝑎
4		nffr.r	. . . . . 6 ⊢ Ⅎ𝑥𝑅
5		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑥𝑏
6	3, 4, 5	nfbr 5121	. . . . 5 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
7	6, 2	nfrabw 3318	. . . 4 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏}
8	7	nfel1 2923	. . 3 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V
9	2, 8	nfralw 3151	. 2 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 {𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V
10	1, 9	nfxfr 1855	1 ⊢ Ⅎ𝑥 𝑅 Se 𝐴

Syntax hints:  Ⅎwnf 1786   ∈ wcel 2106  Ⅎwnfc 2887  ∀wral 3064  {crab 3068  Vcvv 3432   class class class wbr 5074   Se wse 5542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-se 5545

This theorem is referenced by:  nfoi  9273