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Theorem csbwrecsg 8346
Description: Move class substitution in and out of the well-founded recursive function generator. (Contributed by ML, 25-Oct-2020.) (Revised by Scott Fenton, 18-Nov-2024.)
Assertion
Ref Expression
csbwrecsg (𝐴𝑉𝐴 / 𝑥wrecs(𝑅, 𝐷, 𝐹) = wrecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝐹))

Proof of Theorem csbwrecsg
StepHypRef Expression
1 csbfrecsg 8309 . . 3 (𝐴𝑉𝐴 / 𝑥frecs(𝑅, 𝐷, (𝐹 ∘ 2nd )) = frecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥(𝐹 ∘ 2nd )))
2 eqid 2737 . . . 4 𝐴 / 𝑥𝑅 = 𝐴 / 𝑥𝑅
3 eqid 2737 . . . 4 𝐴 / 𝑥𝐷 = 𝐴 / 𝑥𝐷
4 csbcog 6317 . . . . 5 (𝐴𝑉𝐴 / 𝑥(𝐹 ∘ 2nd ) = (𝐴 / 𝑥𝐹𝐴 / 𝑥2nd ))
5 csbconstg 3918 . . . . . 6 (𝐴𝑉𝐴 / 𝑥2nd = 2nd )
65coeq2d 5873 . . . . 5 (𝐴𝑉 → (𝐴 / 𝑥𝐹𝐴 / 𝑥2nd ) = (𝐴 / 𝑥𝐹 ∘ 2nd ))
74, 6eqtrd 2777 . . . 4 (𝐴𝑉𝐴 / 𝑥(𝐹 ∘ 2nd ) = (𝐴 / 𝑥𝐹 ∘ 2nd ))
8 frecseq123 8307 . . . 4 ((𝐴 / 𝑥𝑅 = 𝐴 / 𝑥𝑅𝐴 / 𝑥𝐷 = 𝐴 / 𝑥𝐷𝐴 / 𝑥(𝐹 ∘ 2nd ) = (𝐴 / 𝑥𝐹 ∘ 2nd )) → frecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥(𝐹 ∘ 2nd )) = frecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, (𝐴 / 𝑥𝐹 ∘ 2nd )))
92, 3, 7, 8mp3an12i 1467 . . 3 (𝐴𝑉 → frecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥(𝐹 ∘ 2nd )) = frecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, (𝐴 / 𝑥𝐹 ∘ 2nd )))
101, 9eqtrd 2777 . 2 (𝐴𝑉𝐴 / 𝑥frecs(𝑅, 𝐷, (𝐹 ∘ 2nd )) = frecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, (𝐴 / 𝑥𝐹 ∘ 2nd )))
11 df-wrecs 8337 . . 3 wrecs(𝑅, 𝐷, 𝐹) = frecs(𝑅, 𝐷, (𝐹 ∘ 2nd ))
1211csbeq2i 3907 . 2 𝐴 / 𝑥wrecs(𝑅, 𝐷, 𝐹) = 𝐴 / 𝑥frecs(𝑅, 𝐷, (𝐹 ∘ 2nd ))
13 df-wrecs 8337 . 2 wrecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝐹) = frecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, (𝐴 / 𝑥𝐹 ∘ 2nd ))
1410, 12, 133eqtr4g 2802 1 (𝐴𝑉𝐴 / 𝑥wrecs(𝑅, 𝐷, 𝐹) = wrecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  csb 3899  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  ax-sep 5296  ax-nul 5306  ax-pr 5432
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-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  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:  csbrecsg  37329
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