Theorem csbrdgg 37317

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 37316	. . 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 5518	. . . . 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 4546	. . . . . . 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 3826	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑔 = ∅ ↔ 𝑔 = ∅))
5		csbif 4546	. . . . . . . . 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 3826	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Lim dom 𝑔 ↔ Lim dom 𝑔))
7		csbconstg 3881	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪ ran 𝑔 = ∪ ran 𝑔)
8		csbfv12 6906	. . . . . . . . . . 11 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘(𝑔‘∪ dom 𝑔)) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌(𝑔‘∪ dom 𝑔))
9		csbconstg 3881	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑔‘∪ dom 𝑔) = (𝑔‘∪ dom 𝑔))
10	9	fveq2d 6862	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌(𝑔‘∪ dom 𝑔)) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔)))
11	8, 10	eqtrid 2776	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹‘(𝑔‘∪ dom 𝑔)) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔)))
12	6, 7, 11	ifbieq12d 4517	. . . . . . . . 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 2776	. . . . . . . 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 4515	. . . . . . 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 2776	. . . . . 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 5201	. . . . 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 2764	. . . 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 8342	. . . 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 2764	. 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 8378	. . 3 ⊢ rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
22	21	csbeq2i 3870	. 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 8378	. 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 2789	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌rec(𝐹, 𝐼) = rec(⦋𝐴 / 𝑥⦌𝐹, ⦋𝐴 / 𝑥⦌𝐼))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3447  [wsbc 3753  ⦋csb 3862  ∅c0 4296  ifcif 4488  ∪ cuni 4871   ↦ cmpt 5188  dom cdm 5638  ran crn 5639  Lim wlim 6333  ‘cfv 6511  recscrecs 8339  reccrdg 8377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-xp 5644  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-iota 6464  df-fv 6519  df-ov 7390  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378

This theorem is referenced by:  csbfinxpg  37376