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Theorem csbrdgg 36841
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 36840 . . 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 5564 . . . . 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 4589 . . . . . . 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 3857 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑔 = ∅ ↔ 𝑔 = ∅))
5 csbif 4589 . . . . . . . . 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 3857 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥]Lim dom 𝑔 ↔ Lim dom 𝑔))
7 csbconstg 3913 . . . . . . . . . 10 (𝐴𝑉𝐴 / 𝑥 ran 𝑔 = ran 𝑔)
8 csbfv12 6950 . . . . . . . . . . 11 𝐴 / 𝑥(𝐹‘(𝑔 dom 𝑔)) = (𝐴 / 𝑥𝐹𝐴 / 𝑥(𝑔 dom 𝑔))
9 csbconstg 3913 . . . . . . . . . . . 12 (𝐴𝑉𝐴 / 𝑥(𝑔 dom 𝑔) = (𝑔 dom 𝑔))
109fveq2d 6906 . . . . . . . . . . 11 (𝐴𝑉 → (𝐴 / 𝑥𝐹𝐴 / 𝑥(𝑔 dom 𝑔)) = (𝐴 / 𝑥𝐹‘(𝑔 dom 𝑔)))
118, 10eqtrid 2780 . . . . . . . . . 10 (𝐴𝑉𝐴 / 𝑥(𝐹‘(𝑔 dom 𝑔)) = (𝐴 / 𝑥𝐹‘(𝑔 dom 𝑔)))
126, 7, 11ifbieq12d 4560 . . . . . . . . 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 2780 . . . . . . . 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 4558 . . . . . . 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 2780 . . . . . 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 5254 . . . . 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 2768 . . . 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 8401 . . . 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 2768 . 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 8437 . . 3 rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
2221csbeq2i 3902 . 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 8437 . 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 2793 1 (𝐴𝑉𝐴 / 𝑥rec(𝐹, 𝐼) = rec(𝐴 / 𝑥𝐹, 𝐴 / 𝑥𝐼))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3473  [wsbc 3778  csb 3894  c0 4326  ifcif 4532   cuni 4912  cmpt 5235  dom cdm 5682  ran crn 5683  Lim wlim 6375  cfv 6553  recscrecs 8397  reccrdg 8436
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-xp 5688  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-iota 6505  df-fv 6561  df-ov 7429  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437
This theorem is referenced by:  csbfinxpg  36900
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