Theorem nfwrecs 8258

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 8256	. 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 2899	. . . 4 ⊢ Ⅎ𝑥2nd
6	4, 5	nfco 5815	. . 3 ⊢ Ⅎ𝑥(𝐹 ∘ 2nd )
7	2, 3, 6	nffrecs 8227	. 2 ⊢ Ⅎ𝑥frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
8	1, 7	nfcxfr 2897	1 ⊢ Ⅎ𝑥wrecs(𝑅, 𝐴, 𝐹)

Syntax hints:  Ⅎwnfc 2884   ∘ ccom 5629  2^nd c2nd 7935  frecscfrecs 8224  wrecscwrecs 8255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5631  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-iota 6449  df-fv 6501  df-ov 7364  df-frecs 8225  df-wrecs 8256

This theorem is referenced by:  nfrecs  8308