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Theorem nfwrecs 8300
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 8296 . 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 2903 . . . 4 𝑥2nd
64, 5nfco 5865 . . 3 𝑥(𝐹 ∘ 2nd )
72, 3, 6nffrecs 8267 . 2 𝑥frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
81, 7nfcxfr 2901 1 𝑥wrecs(𝑅, 𝐴, 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2883  ccom 5680  2nd c2nd 7973  frecscfrecs 8264  wrecscwrecs 8295
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-iota 6495  df-fv 6551  df-ov 7411  df-frecs 8265  df-wrecs 8296
This theorem is referenced by:  nfrecs  8374
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