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Theorem csbrdgg 32799
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 32798 . . 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 4976 . . . . 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 4115 . . . . . . 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 3490 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑔 = ∅ ↔ 𝑔 = ∅))
5 csbif 4115 . . . . . . . . 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 3490 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥]Lim dom 𝑔 ↔ Lim dom 𝑔))
7 csbconstg 3532 . . . . . . . . . 10 (𝐴𝑉𝐴 / 𝑥 ran 𝑔 = ran 𝑔)
8 csbfv12 6189 . . . . . . . . . . 11 𝐴 / 𝑥(𝐹‘(𝑔 dom 𝑔)) = (𝐴 / 𝑥𝐹𝐴 / 𝑥(𝑔 dom 𝑔))
9 csbconstg 3532 . . . . . . . . . . . 12 (𝐴𝑉𝐴 / 𝑥(𝑔 dom 𝑔) = (𝑔 dom 𝑔))
109fveq2d 6154 . . . . . . . . . . 11 (𝐴𝑉 → (𝐴 / 𝑥𝐹𝐴 / 𝑥(𝑔 dom 𝑔)) = (𝐴 / 𝑥𝐹‘(𝑔 dom 𝑔)))
118, 10syl5eq 2672 . . . . . . . . . 10 (𝐴𝑉𝐴 / 𝑥(𝐹‘(𝑔 dom 𝑔)) = (𝐴 / 𝑥𝐹‘(𝑔 dom 𝑔)))
126, 7, 11ifbieq12d 4090 . . . . . . . . 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, 12syl5eq 2672 . . . . . . . 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 4088 . . . . . . 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, 14syl5eq 2672 . . . . . 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 4710 . . . . 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 2660 . . . 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 7416 . . . 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 2660 . 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 7452 . . 3 rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
2221csbeq2i 3970 . 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 7452 . 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 2685 1 (𝐴𝑉𝐴 / 𝑥rec(𝐹, 𝐼) = rec(𝐴 / 𝑥𝐹, 𝐴 / 𝑥𝐼))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1992  Vcvv 3191  [wsbc 3422  csb 3519  c0 3896  ifcif 4063   cuni 4407  cmpt 4678  dom cdm 5079  ran crn 5080  Lim wlim 5686  cfv 5850  recscrecs 7413  reccrdg 7451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-xp 5085  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-iota 5813  df-fv 5858  df-wrecs 7353  df-recs 7414  df-rdg 7452
This theorem is referenced by:  csbfinxpg  32849
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