Theorem csbrdgg 34746

Description: Move class substitution in and out of the recursive function generator. (Contributed by ML, 25-Oct-2020.)

Assertion

Ref	Expression
csbrdgg	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌rec(𝐹, 𝐼) = rec(⦋𝐴 / 𝑥⦌𝐹, ⦋𝐴 / 𝑥⦌𝐼))

Proof of Theorem csbrdgg

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbrecsg 34745	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌recs((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 5410	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) = (𝑔 ∈ V ↦ ⦋𝐴 / 𝑥⦌if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
3		csbif 4480	. . . . . . 7 ⊢ ⦋𝐴 / 𝑥⦌if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))) = if([𝐴 / 𝑥]𝑔 = ∅, ⦋𝐴 / 𝑥⦌𝐼, ⦋𝐴 / 𝑥⦌if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))
4		sbcg 3793	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑔 = ∅ ↔ 𝑔 = ∅))
5		csbif 4480	. . . . . . . . 9 ⊢ ⦋𝐴 / 𝑥⦌if(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 3793	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Lim dom 𝑔 ↔ Lim dom 𝑔))
7		csbconstg 3847	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪ ran 𝑔 = ∪ ran 𝑔)
8		csbfv12 6688	. . . . . . . . . . 11 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘(𝑔‘∪ dom 𝑔)) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌(𝑔‘∪ dom 𝑔))
9		csbconstg 3847	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑔‘∪ dom 𝑔) = (𝑔‘∪ dom 𝑔))
10	9	fveq2d 6649	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌(𝑔‘∪ dom 𝑔)) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔)))
11	8, 10	syl5eq 2845	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹‘(𝑔‘∪ dom 𝑔)) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔)))
12	6, 7, 11	ifbieq12d 4452	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → if([𝐴 / 𝑥]Lim dom 𝑔, ⦋𝐴 / 𝑥⦌∪ ran 𝑔, ⦋𝐴 / 𝑥⦌(𝐹‘(𝑔‘∪ dom 𝑔))) = if(Lim dom 𝑔, ∪ ran 𝑔, (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔))))
13	5, 12	syl5eq 2845	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))) = if(Lim dom 𝑔, ∪ ran 𝑔, (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔))))
14	4, 13	ifbieq2d 4450	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → if([𝐴 / 𝑥]𝑔 = ∅, ⦋𝐴 / 𝑥⦌𝐼, ⦋𝐴 / 𝑥⦌if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))) = if(𝑔 = ∅, ⦋𝐴 / 𝑥⦌𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔)))))
15	3, 14	syl5eq 2845	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))) = if(𝑔 = ∅, ⦋𝐴 / 𝑥⦌𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔)))))
16	15	mpteq2dv 5126	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑔 ∈ V ↦ ⦋𝐴 / 𝑥⦌if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, ⦋𝐴 / 𝑥⦌𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔))))))
17	2, 16	eqtrd 2833	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, ⦋𝐴 / 𝑥⦌𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔))))))
18		recseq 7993	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, ⦋𝐴 / 𝑥⦌𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (⦋𝐴 / 𝑥⦌𝐹‘(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 / 𝑥⦌𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔)))))))
19	17, 18	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → recs(⦋𝐴 / 𝑥⦌(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, ⦋𝐴 / 𝑥⦌𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔)))))))
20	1, 19	eqtrd 2833	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, ⦋𝐴 / 𝑥⦌𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔)))))))
21		df-rdg 8029	. . 3 ⊢ rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
22	21	csbeq2i 3836	. 2 ⊢ ⦋𝐴 / 𝑥⦌rec(𝐹, 𝐼) = ⦋𝐴 / 𝑥⦌recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
23		df-rdg 8029	. 2 ⊢ rec(⦋𝐴 / 𝑥⦌𝐹, ⦋𝐴 / 𝑥⦌𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, ⦋𝐴 / 𝑥⦌𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔))))))
24	20, 22, 23	3eqtr4g 2858	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌rec(𝐹, 𝐼) = rec(⦋𝐴 / 𝑥⦌𝐹, ⦋𝐴 / 𝑥⦌𝐼))

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3441  [wsbc 3720  ⦋csb 3828  ∅c0 4243  ifcif 4425  ∪ cuni 4800   ↦ cmpt 5110  dom cdm 5519  ran crn 5520  Lim wlim 6160  ‘cfv 6324  recscrecs 7990  reccrdg 8028

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-iota 6283  df-fv 6332  df-wrecs 7930  df-recs 7991  df-rdg 8029

This theorem is referenced by:  csbfinxpg  34805