Theorem csbrdgg 37771

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 37770	. . 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 5522	. . . . 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 4532	. . . . . . 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 3811	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑔 = ∅ ↔ 𝑔 = ∅))
5		csbif 4532	. . . . . . . . 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 3811	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Lim dom 𝑔 ↔ Lim dom 𝑔))
7		csbconstg 3866	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪ ran 𝑔 = ∪ ran 𝑔)
8		csbfv12 6901	. . . . . . . . . . 11 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘(𝑔‘∪ dom 𝑔)) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌(𝑔‘∪ dom 𝑔))
9		csbconstg 3866	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑔‘∪ dom 𝑔) = (𝑔‘∪ dom 𝑔))
10	9	fveq2d 6860	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌(𝑔‘∪ dom 𝑔)) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔)))
11	8, 10	eqtrid 2803	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹‘(𝑔‘∪ dom 𝑔)) = (⦋𝐴 / 𝑥⦌𝐹‘(𝑔‘∪ dom 𝑔)))
12	6, 7, 11	ifbieq12d 4503	. . . . . . . . 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 2803	. . . . . . . 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 4501	. . . . . . 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 2803	. . . . . 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 5188	. . . . 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 2791	. . . 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 8332	. . . 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 2791	. 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 8369	. . 3 ⊢ rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
22	21	csbeq2i 3855	. 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 8369	. 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 2816	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌rec(𝐹, 𝐼) = rec(⦋𝐴 / 𝑥⦌𝐹, ⦋𝐴 / 𝑥⦌𝐼))

Syntax hints:   → wi 4   = wceq 1554   ∈ wcel 2136  Vcvv 3448  [wsbc 3739  ⦋csb 3847  ∅c0 4280  ifcif 4474  ∪ cuni 4859   ↦ cmpt 5175  dom cdm 5640  ran crn 5641  Lim wlim 6336  ‘cfv 6510  recscrecs 8329  reccrdg 8368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-mpt 5176  df-xp 5646  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6277  df-iota 6466  df-fv 6518  df-ov 7388  df-frecs 8250  df-wrecs 8281  df-recs 8330  df-rdg 8369

This theorem is referenced by:  csbfinxpg  37830