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Theorem csbwrecsg 8307
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 8270 . . 3 (𝐴𝑉𝐴 / 𝑥frecs(𝑅, 𝐷, (𝐹 ∘ 2nd )) = frecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥(𝐹 ∘ 2nd )))
2 eqid 2726 . . . 4 𝐴 / 𝑥𝑅 = 𝐴 / 𝑥𝑅
3 eqid 2726 . . . 4 𝐴 / 𝑥𝐷 = 𝐴 / 𝑥𝐷
4 csbcog 6290 . . . . 5 (𝐴𝑉𝐴 / 𝑥(𝐹 ∘ 2nd ) = (𝐴 / 𝑥𝐹𝐴 / 𝑥2nd ))
5 csbconstg 3907 . . . . . 6 (𝐴𝑉𝐴 / 𝑥2nd = 2nd )
65coeq2d 5856 . . . . 5 (𝐴𝑉 → (𝐴 / 𝑥𝐹𝐴 / 𝑥2nd ) = (𝐴 / 𝑥𝐹 ∘ 2nd ))
74, 6eqtrd 2766 . . . 4 (𝐴𝑉𝐴 / 𝑥(𝐹 ∘ 2nd ) = (𝐴 / 𝑥𝐹 ∘ 2nd ))
8 frecseq123 8268 . . . 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 1461 . . 3 (𝐴𝑉 → frecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥(𝐹 ∘ 2nd )) = frecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, (𝐴 / 𝑥𝐹 ∘ 2nd )))
101, 9eqtrd 2766 . 2 (𝐴𝑉𝐴 / 𝑥frecs(𝑅, 𝐷, (𝐹 ∘ 2nd )) = frecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, (𝐴 / 𝑥𝐹 ∘ 2nd )))
11 df-wrecs 8298 . . 3 wrecs(𝑅, 𝐷, 𝐹) = frecs(𝑅, 𝐷, (𝐹 ∘ 2nd ))
1211csbeq2i 3896 . 2 𝐴 / 𝑥wrecs(𝑅, 𝐷, 𝐹) = 𝐴 / 𝑥frecs(𝑅, 𝐷, (𝐹 ∘ 2nd ))
13 df-wrecs 8298 . 2 wrecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝐹) = frecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, (𝐴 / 𝑥𝐹 ∘ 2nd ))
1410, 12, 133eqtr4g 2791 1 (𝐴𝑉𝐴 / 𝑥wrecs(𝑅, 𝐷, 𝐹) = wrecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  csb 3888  ccom 5673  2nd c2nd 7973  frecscfrecs 8266  wrecscwrecs 8297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-iota 6489  df-fv 6545  df-ov 7408  df-frecs 8267  df-wrecs 8298
This theorem is referenced by:  csbrecsg  36716
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