Theorem nfse 4343

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 4335	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑏 ∈ 𝐴 {𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V)
2		nfse.a	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcv 2319	. . . . . 6 ⊢ Ⅎ𝑥𝑎
4		nfse.r	. . . . . 6 ⊢ Ⅎ𝑥𝑅
5		nfcv 2319	. . . . . 6 ⊢ Ⅎ𝑥𝑏
6	3, 4, 5	nfbr 4051	. . . . 5 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
7	6, 2	nfrabxy 2658	. . . 4 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏}
8	7	nfel1 2330	. . 3 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V
9	2, 8	nfralxy 2515	. 2 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 {𝑎 ∈ 𝐴 ∣ 𝑎𝑅𝑏} ∈ V
10	1, 9	nfxfr 1474	1 ⊢ Ⅎ𝑥 𝑅 Se 𝐴

Syntax hints:  Ⅎwnf 1460   ∈ wcel 2148  Ⅎwnfc 2306  ∀wral 2455  {crab 2459  Vcvv 2739   class class class wbr 4005   Se wse 4331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-se 4335

This theorem is referenced by: (None)