Theorem csbrdgg 36296

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 36295	. . 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 5558	. . . . 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 4585	. . . . . . 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 3856	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑔 = ∅ ↔ 𝑔 = ∅))
5		csbif 4585	. . . . . . . . 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 3856	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Lim dom 𝑔 ↔ Lim dom 𝑔))
7		csbconstg 3912	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪ ran 𝑔 = ∪ ran 𝑔)
8		csbfv12 6939	. . . . . . . . . . 11 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘(𝑔‘∪ dom 𝑔)) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌(𝑔‘∪ dom 𝑔))
9		csbconstg 3912	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑔‘∪ dom 𝑔) = (𝑔‘∪ dom 𝑔))
10	9	fveq2d 6895	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌(𝑔‘∪ dom 𝑔)) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔)))
11	8, 10	eqtrid 2784	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹‘(𝑔‘∪ dom 𝑔)) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔)))
12	6, 7, 11	ifbieq12d 4556	. . . . . . . . 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 2784	. . . . . . . 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 4554	. . . . . . 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 2784	. . . . . 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 5250	. . . . 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 2772	. . . 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 8376	. . . 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 2772	. 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 8412	. . 3 ⊢ rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
22	21	csbeq2i 3901	. 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 8412	. 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 2797	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌rec(𝐹, 𝐼) = rec(⦋𝐴 / 𝑥⦌𝐹, ⦋𝐴 / 𝑥⦌𝐼))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3474  [wsbc 3777  ⦋csb 3893  ∅c0 4322  ifcif 4528  ∪ cuni 4908   ↦ cmpt 5231  dom cdm 5676  ran crn 5677  Lim wlim 6365  ‘cfv 6543  recscrecs 8372  reccrdg 8411

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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5682  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-iota 6495  df-fv 6551  df-ov 7414  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412

This theorem is referenced by:  csbfinxpg  36355