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Theorem nfwrecs 8341
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 𝑥𝑅
nfwrecs.2 𝑥𝐴
nfwrecs.3 𝑥𝐹
Assertion
Ref Expression
nfwrecs 𝑥wrecs(𝑅, 𝐴, 𝐹)

Proof of Theorem nfwrecs
StepHypRef Expression
1 df-wrecs 8337 . 2 wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
2 nfwrecs.1 . . 3 𝑥𝑅
3 nfwrecs.2 . . 3 𝑥𝐴
4 nfwrecs.3 . . . 4 𝑥𝐹
5 nfcv 2905 . . . 4 𝑥2nd
64, 5nfco 5876 . . 3 𝑥(𝐹 ∘ 2nd )
72, 3, 6nffrecs 8308 . 2 𝑥frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
81, 7nfcxfr 2903 1 𝑥wrecs(𝑅, 𝐴, 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2890  ccom 5689  2nd c2nd 8013  frecscfrecs 8305  wrecscwrecs 8336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fv 6569  df-ov 7434  df-frecs 8306  df-wrecs 8337
This theorem is referenced by:  nfrecs  8415
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