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Theorem csbrdgg 35800
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 35799 . . 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 5515 . . . . 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 4543 . . . . . . 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 3818 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑔 = ∅ ↔ 𝑔 = ∅))
5 csbif 4543 . . . . . . . . 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 3818 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥]Lim dom 𝑔 ↔ Lim dom 𝑔))
7 csbconstg 3874 . . . . . . . . . 10 (𝐴𝑉𝐴 / 𝑥 ran 𝑔 = ran 𝑔)
8 csbfv12 6890 . . . . . . . . . . 11 𝐴 / 𝑥(𝐹‘(𝑔 dom 𝑔)) = (𝐴 / 𝑥𝐹𝐴 / 𝑥(𝑔 dom 𝑔))
9 csbconstg 3874 . . . . . . . . . . . 12 (𝐴𝑉𝐴 / 𝑥(𝑔 dom 𝑔) = (𝑔 dom 𝑔))
109fveq2d 6846 . . . . . . . . . . 11 (𝐴𝑉 → (𝐴 / 𝑥𝐹𝐴 / 𝑥(𝑔 dom 𝑔)) = (𝐴 / 𝑥𝐹‘(𝑔 dom 𝑔)))
118, 10eqtrid 2788 . . . . . . . . . 10 (𝐴𝑉𝐴 / 𝑥(𝐹‘(𝑔 dom 𝑔)) = (𝐴 / 𝑥𝐹‘(𝑔 dom 𝑔)))
126, 7, 11ifbieq12d 4514 . . . . . . . . 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 2788 . . . . . . . 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 4512 . . . . . . 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 2788 . . . . . 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 5207 . . . . 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 2776 . . . 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 8320 . . . 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 2776 . 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 8356 . . 3 rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
2221csbeq2i 3863 . 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 8356 . 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 2801 1 (𝐴𝑉𝐴 / 𝑥rec(𝐹, 𝐼) = rec(𝐴 / 𝑥𝐹, 𝐴 / 𝑥𝐼))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  Vcvv 3445  [wsbc 3739  csb 3855  c0 4282  ifcif 4486   cuni 4865  cmpt 5188  dom cdm 5633  ran crn 5634  Lim wlim 6318  cfv 6496  recscrecs 8316  reccrdg 8355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-xp 5639  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-iota 6448  df-fv 6504  df-ov 7360  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356
This theorem is referenced by:  csbfinxpg  35859
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