Theorem csbrdgg 37858

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 37857	. . 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 5541	. . . . 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 4547	. . . . . . 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 3825	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑔 = ∅ ↔ 𝑔 = ∅))
5		csbif 4547	. . . . . . . . 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 3825	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Lim dom 𝑔 ↔ Lim dom 𝑔))
7		csbconstg 3880	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪ ran 𝑔 = ∪ ran 𝑔)
8		csbfv12 6924	. . . . . . . . . . 11 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘(𝑔‘∪ dom 𝑔)) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌(𝑔‘∪ dom 𝑔))
9		csbconstg 3880	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑔‘∪ dom 𝑔) = (𝑔‘∪ dom 𝑔))
10	9	fveq2d 6883	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌(𝑔‘∪ dom 𝑔)) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔)))
11	8, 10	eqtrid 2816	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹‘(𝑔‘∪ dom 𝑔)) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔)))
12	6, 7, 11	ifbieq12d 4518	. . . . . . . . 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	eqtrid 2816	. . . . . . . 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 4516	. . . . . . 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	eqtrid 2816	. . . . . 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 5206	. . . . 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 2804	. . . 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 8356	. . . 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 18	. . 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 2804	. 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 8393	. . 3 ⊢ rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
22	21	csbeq2i 3869	. 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 8393	. 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 2829	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌rec(𝐹, 𝐼) = rec(⦋𝐴 / 𝑥⦌𝐹, ⦋𝐴 / 𝑥⦌𝐼))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  Vcvv 3463  [wsbc 3753  ⦋csb 3861  ∅c0 4294  ifcif 4489  ∪ cuni 4873   ↦ cmpt 5193  dom cdm 5659  ran crn 5660  Lim wlim 6358  ‘cfv 6533  recscrecs 8353  reccrdg 8392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-xp 5665  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-iota 6489  df-fv 6541  df-ov 7411  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393

This theorem is referenced by:  csbfinxpg  37917