Theorem nfrel 4705

Description: Bound-variable hypothesis builder for a relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypothesis

Ref	Expression
nfrel.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfrel	⊢ Ⅎ𝑥Rel 𝐴

Proof of Theorem nfrel

Step	Hyp	Ref	Expression
1		df-rel 4627	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
2		nfrel.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcv 2317	. . 3 ⊢ Ⅎ𝑥(V × V)
4	2, 3	nfss 3146	. 2 ⊢ Ⅎ𝑥 𝐴 ⊆ (V × V)
5	1, 4	nfxfr 1472	1 ⊢ Ⅎ𝑥Rel 𝐴

Syntax hints:  Ⅎwnf 1458  Ⅎwnfc 2304  Vcvv 2735   ⊆ wss 3127   × cxp 4618  Rel wrel 4625

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-in 3133  df-ss 3140  df-rel 4627

This theorem is referenced by:  nffun  5231