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Theorem csbwrecsg 8325
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 8288 . . 3 (𝐴𝑉𝐴 / 𝑥frecs(𝑅, 𝐷, (𝐹 ∘ 2nd )) = frecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥(𝐹 ∘ 2nd )))
2 eqid 2725 . . . 4 𝐴 / 𝑥𝑅 = 𝐴 / 𝑥𝑅
3 eqid 2725 . . . 4 𝐴 / 𝑥𝐷 = 𝐴 / 𝑥𝐷
4 csbcog 6296 . . . . 5 (𝐴𝑉𝐴 / 𝑥(𝐹 ∘ 2nd ) = (𝐴 / 𝑥𝐹𝐴 / 𝑥2nd ))
5 csbconstg 3903 . . . . . 6 (𝐴𝑉𝐴 / 𝑥2nd = 2nd )
65coeq2d 5859 . . . . 5 (𝐴𝑉 → (𝐴 / 𝑥𝐹𝐴 / 𝑥2nd ) = (𝐴 / 𝑥𝐹 ∘ 2nd ))
74, 6eqtrd 2765 . . . 4 (𝐴𝑉𝐴 / 𝑥(𝐹 ∘ 2nd ) = (𝐴 / 𝑥𝐹 ∘ 2nd ))
8 frecseq123 8286 . . . 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 2765 . 2 (𝐴𝑉𝐴 / 𝑥frecs(𝑅, 𝐷, (𝐹 ∘ 2nd )) = frecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, (𝐴 / 𝑥𝐹 ∘ 2nd )))
11 df-wrecs 8316 . . 3 wrecs(𝑅, 𝐷, 𝐹) = frecs(𝑅, 𝐷, (𝐹 ∘ 2nd ))
1211csbeq2i 3892 . 2 𝐴 / 𝑥wrecs(𝑅, 𝐷, 𝐹) = 𝐴 / 𝑥frecs(𝑅, 𝐷, (𝐹 ∘ 2nd ))
13 df-wrecs 8316 . 2 wrecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝐹) = frecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, (𝐴 / 𝑥𝐹 ∘ 2nd ))
1410, 12, 133eqtr4g 2790 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 3884  ccom 5676  2nd c2nd 7990  frecscfrecs 8284  wrecscwrecs 8315
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-xp 5678  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-iota 6495  df-fv 6551  df-ov 7419  df-frecs 8285  df-wrecs 8316
This theorem is referenced by:  csbrecsg  36864
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