Theorem nfwrecs 8301

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 8297	. 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 2904	. . . 4 ⊢ Ⅎ𝑥2nd
6	4, 5	nfco 5866	. . 3 ⊢ Ⅎ𝑥(𝐹 ∘ 2nd )
7	2, 3, 6	nffrecs 8268	. 2 ⊢ Ⅎ𝑥frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
8	1, 7	nfcxfr 2902	1 ⊢ Ⅎ𝑥wrecs(𝑅, 𝐴, 𝐹)

Syntax hints:  Ⅎwnfc 2884   ∘ ccom 5681  2^nd c2nd 7974  frecscfrecs 8265  wrecscwrecs 8296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-iota 6496  df-fv 6552  df-ov 7412  df-frecs 8266  df-wrecs 8297

This theorem is referenced by:  nfrecs  8375