Theorem nfwrecs 8103

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 8099	. 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 2906	. . . 4 ⊢ Ⅎ𝑥2nd
6	4, 5	nfco 5763	. . 3 ⊢ Ⅎ𝑥(𝐹 ∘ 2nd )
7	2, 3, 6	nffrecs 8070	. 2 ⊢ Ⅎ𝑥frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
8	1, 7	nfcxfr 2904	1 ⊢ Ⅎ𝑥wrecs(𝑅, 𝐴, 𝐹)

Syntax hints:  Ⅎwnfc 2886   ∘ ccom 5584  2^nd c2nd 7803  frecscfrecs 8067  wrecscwrecs 8098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-iota 6376  df-fv 6426  df-ov 7258  df-frecs 8068  df-wrecs 8099

This theorem is referenced by:  nfrecs  8177