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Theorem csbrdgg 37330
Description: Move class substitution in and out of the recursive function generator. (Contributed by ML, 25-Oct-2020.)
Assertion
Ref Expression
csbrdgg (𝐴𝑉𝐴 / 𝑥rec(𝐹, 𝐼) = rec(𝐴 / 𝑥𝐹, 𝐴 / 𝑥𝐼))

Proof of Theorem csbrdgg
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 csbrecsg 37329 . . 3 (𝐴𝑉𝐴 / 𝑥recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))) = recs(𝐴 / 𝑥(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))))
2 csbmpt2 5563 . . . . 5 (𝐴𝑉𝐴 / 𝑥(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))) = (𝑔 ∈ V ↦ 𝐴 / 𝑥if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
3 csbif 4583 . . . . . . 7 𝐴 / 𝑥if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))) = if([𝐴 / 𝑥]𝑔 = ∅, 𝐴 / 𝑥𝐼, 𝐴 / 𝑥if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))
4 sbcg 3863 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑔 = ∅ ↔ 𝑔 = ∅))
5 csbif 4583 . . . . . . . . 9 𝐴 / 𝑥if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))) = if([𝐴 / 𝑥]Lim dom 𝑔, 𝐴 / 𝑥 ran 𝑔, 𝐴 / 𝑥(𝐹‘(𝑔 dom 𝑔)))
6 sbcg 3863 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥]Lim dom 𝑔 ↔ Lim dom 𝑔))
7 csbconstg 3918 . . . . . . . . . 10 (𝐴𝑉𝐴 / 𝑥 ran 𝑔 = ran 𝑔)
8 csbfv12 6954 . . . . . . . . . . 11 𝐴 / 𝑥(𝐹‘(𝑔 dom 𝑔)) = (𝐴 / 𝑥𝐹𝐴 / 𝑥(𝑔 dom 𝑔))
9 csbconstg 3918 . . . . . . . . . . . 12 (𝐴𝑉𝐴 / 𝑥(𝑔 dom 𝑔) = (𝑔 dom 𝑔))
109fveq2d 6910 . . . . . . . . . . 11 (𝐴𝑉 → (𝐴 / 𝑥𝐹𝐴 / 𝑥(𝑔 dom 𝑔)) = (𝐴 / 𝑥𝐹‘(𝑔 dom 𝑔)))
118, 10eqtrid 2789 . . . . . . . . . 10 (𝐴𝑉𝐴 / 𝑥(𝐹‘(𝑔 dom 𝑔)) = (𝐴 / 𝑥𝐹‘(𝑔 dom 𝑔)))
126, 7, 11ifbieq12d 4554 . . . . . . . . 9 (𝐴𝑉 → if([𝐴 / 𝑥]Lim dom 𝑔, 𝐴 / 𝑥 ran 𝑔, 𝐴 / 𝑥(𝐹‘(𝑔 dom 𝑔))) = if(Lim dom 𝑔, ran 𝑔, (𝐴 / 𝑥𝐹‘(𝑔 dom 𝑔))))
135, 12eqtrid 2789 . . . . . . . 8 (𝐴𝑉𝐴 / 𝑥if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))) = if(Lim dom 𝑔, ran 𝑔, (𝐴 / 𝑥𝐹‘(𝑔 dom 𝑔))))
144, 13ifbieq2d 4552 . . . . . . 7 (𝐴𝑉 → if([𝐴 / 𝑥]𝑔 = ∅, 𝐴 / 𝑥𝐼, 𝐴 / 𝑥if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))) = if(𝑔 = ∅, 𝐴 / 𝑥𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐴 / 𝑥𝐹‘(𝑔 dom 𝑔)))))
153, 14eqtrid 2789 . . . . . 6 (𝐴𝑉𝐴 / 𝑥if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))) = if(𝑔 = ∅, 𝐴 / 𝑥𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐴 / 𝑥𝐹‘(𝑔 dom 𝑔)))))
1615mpteq2dv 5244 . . . . 5 (𝐴𝑉 → (𝑔 ∈ V ↦ 𝐴 / 𝑥if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴 / 𝑥𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐴 / 𝑥𝐹‘(𝑔 dom 𝑔))))))
172, 16eqtrd 2777 . . . 4 (𝐴𝑉𝐴 / 𝑥(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴 / 𝑥𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐴 / 𝑥𝐹‘(𝑔 dom 𝑔))))))
18 recseq 8414 . . . 4 (𝐴 / 𝑥(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴 / 𝑥𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐴 / 𝑥𝐹‘(𝑔 dom 𝑔))))) → recs(𝐴 / 𝑥(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴 / 𝑥𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐴 / 𝑥𝐹‘(𝑔 dom 𝑔)))))))
1917, 18syl 17 . . 3 (𝐴𝑉 → recs(𝐴 / 𝑥(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴 / 𝑥𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐴 / 𝑥𝐹‘(𝑔 dom 𝑔)))))))
201, 19eqtrd 2777 . 2 (𝐴𝑉𝐴 / 𝑥recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴 / 𝑥𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐴 / 𝑥𝐹‘(𝑔 dom 𝑔)))))))
21 df-rdg 8450 . . 3 rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
2221csbeq2i 3907 . 2 𝐴 / 𝑥rec(𝐹, 𝐼) = 𝐴 / 𝑥recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
23 df-rdg 8450 . 2 rec(𝐴 / 𝑥𝐹, 𝐴 / 𝑥𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴 / 𝑥𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐴 / 𝑥𝐹‘(𝑔 dom 𝑔))))))
2420, 22, 233eqtr4g 2802 1 (𝐴𝑉𝐴 / 𝑥rec(𝐹, 𝐼) = rec(𝐴 / 𝑥𝐹, 𝐴 / 𝑥𝐼))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  [wsbc 3788  csb 3899  c0 4333  ifcif 4525   cuni 4907  cmpt 5225  dom cdm 5685  ran crn 5686  Lim wlim 6385  cfv 6561  recscrecs 8410  reccrdg 8449
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-mpt 5226  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  df-recs 8411  df-rdg 8450
This theorem is referenced by:  csbfinxpg  37389
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