Theorem nfwrecs 8332

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 8330	. 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 2923	. . . 4 ⊢ Ⅎ𝑥2nd
6	4, 5	nfco 5843	. . 3 ⊢ Ⅎ𝑥(𝐹 ∘ 2nd )
7	2, 3, 6	nffrecs 8301	. 2 ⊢ Ⅎ𝑥frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
8	1, 7	nfcxfr 2921	1 ⊢ Ⅎ𝑥wrecs(𝑅, 𝐴, 𝐹)

This proof depends on syntax axioms:  Ⅎwnfc 2908   ∘ ccom 5655  2^nd c2nd 8000  frecscfrecs 8298  wrecscwrecs 8329

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5657  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-iota 6494  df-fv 6546  df-ov 7423  df-frecs 8299  df-wrecs 8330

This theorem is used by:  nfrecs  8382