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 2922	. . . . . 6 ⊢ Ⅎ𝑥𝑎
4		nffr.r	. . . . . 6 ⊢ Ⅎ𝑥𝑅
5		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑥𝑏
6	3, 4, 5	nfbr 5151	. . . . 5 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
7	6, 2	nfrabw 3447	. . . 4 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏}
8	7	nfel1 2938	. . 3 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V
9	2, 8	nfralw 3309	. 2 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 {𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V
10	1, 9	nfxfr 1885	1 ⊢ Ⅎ𝑥 𝑅 Se 𝐴

This proof depends on syntax axioms:  Ⅎwnf 1815   ∈ wcel 2144  Ⅎwnfc 2907  ∀wral 3076  {crab 3412  Vcvv 3450   class class class wbr 5102   Se wse 5602

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-se 5605

This theorem is used by:  nfoi  9515