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Theorem nfwrecs 8340
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 8336 . 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 5879 . . 3 𝑥(𝐹 ∘ 2nd )
72, 3, 6nffrecs 8307 . 2 𝑥frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
81, 7nfcxfr 2901 1 𝑥wrecs(𝑅, 𝐴, 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2888  ccom 5693  2nd c2nd 8012  frecscfrecs 8304  wrecscwrecs 8335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fv 6571  df-ov 7434  df-frecs 8305  df-wrecs 8336
This theorem is referenced by:  nfrecs  8414
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