Theorem nfse 5625

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 5605	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑏 ∈ 𝐴 {𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V)
2		nffr.a	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑥𝑎
4		nffr.r	. . . . . 6 ⊢ Ⅎ𝑥𝑅
5		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑥𝑏
6	3, 4, 5	nfbr 5152	. . . . 5 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
7	6, 2	nfrabw 3448	. . . 4 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏}
8	7	nfel1 2939	. . 3 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V
9	2, 8	nfralw 3310	. 2 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 {𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V
10	1, 9	nfxfr 1886	1 ⊢ Ⅎ𝑥 𝑅 Se 𝐴

This proof depends on syntax axioms:  Ⅎwnf 1816   ∈ wcel 2145  Ⅎwnfc 2908  ∀wral 3077  {crab 3413  Vcvv 3451   class class class wbr 5103   Se wse 5602

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-se 5605

This theorem is used by:  nfoi  9508