Theorem nfwrecs 8244

Description: Bound-variable hypothesis builder for the well-ordered recursive function generator. (Contributed by Scott Fenton, 9-Jun-2018.) (Proof shortened by Scott Fenton, 17-Nov-2024.)

Hypotheses

Ref	Expression
nfwrecs.1	⊢ Ⅎ𝑥𝑅
nfwrecs.2	⊢ Ⅎ𝑥𝐴
nfwrecs.3	⊢ Ⅎ𝑥𝐹

Assertion

Ref	Expression
nfwrecs	⊢ Ⅎ𝑥wrecs(𝑅, 𝐴, 𝐹)

Proof of Theorem nfwrecs

Step	Hyp	Ref	Expression
1		df-wrecs 8242	. 2 ⊢ wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
2		nfwrecs.1	. . 3 ⊢ Ⅎ𝑥𝑅
3		nfwrecs.2	. . 3 ⊢ Ⅎ𝑥𝐴
4		nfwrecs.3	. . . 4 ⊢ Ⅎ𝑥𝐹
5		nfcv 2894	. . . 4 ⊢ Ⅎ𝑥2nd
6	4, 5	nfco 5804	. . 3 ⊢ Ⅎ𝑥(𝐹 ∘ 2nd )
7	2, 3, 6	nffrecs 8213	. 2 ⊢ Ⅎ𝑥frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
8	1, 7	nfcxfr 2892	1 ⊢ Ⅎ𝑥wrecs(𝑅, 𝐴, 𝐹)

Syntax hints:  Ⅎwnfc 2879   ∘ ccom 5618  2^nd c2nd 7920  frecscfrecs 8210  wrecscwrecs 8241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-iota 6437  df-fv 6489  df-ov 7349  df-frecs 8211  df-wrecs 8242

This theorem is referenced by:  nfrecs  8294