Theorem csbwrecsg 8268

Description: Move class substitution in and out of the well-founded recursive function generator. (Contributed by ML, 25-Oct-2020.) (Revised by Scott Fenton, 18-Nov-2024.)

Assertion

Ref	Expression
csbwrecsg	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌wrecs(𝑅, 𝐷, 𝐹) = wrecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝐹))

Proof of Theorem csbwrecsg

Step	Hyp	Ref	Expression
1		csbfrecsg 8234	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌frecs(𝑅, 𝐷, (𝐹 ∘ 2nd )) = frecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌(𝐹 ∘ 2nd )))
2		eqid 2736	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌𝑅 = ⦋𝐴 / 𝑥⦌𝑅
3		eqid 2736	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌𝐷 = ⦋𝐴 / 𝑥⦌𝐷
4		csbcog 6261	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹 ∘ 2nd ) = (⦋𝐴 / 𝑥⦌𝐹 ∘ ⦋𝐴 / 𝑥⦌2nd ))
5		csbconstg 3856	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌2nd = 2nd )
6	5	coeq2d 5817	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐹 ∘ ⦋𝐴 / 𝑥⦌2nd ) = (⦋𝐴 / 𝑥⦌𝐹 ∘ 2nd ))
7	4, 6	eqtrd 2771	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹 ∘ 2nd ) = (⦋𝐴 / 𝑥⦌𝐹 ∘ 2nd ))
8		frecseq123 8232	. . . 4 ⊢ ((⦋𝐴 / 𝑥⦌𝑅 = ⦋𝐴 / 𝑥⦌𝑅 ∧ ⦋𝐴 / 𝑥⦌𝐷 = ⦋𝐴 / 𝑥⦌𝐷 ∧ ⦋𝐴 / 𝑥⦌(𝐹 ∘ 2nd ) = (⦋𝐴 / 𝑥⦌𝐹 ∘ 2nd )) → frecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌(𝐹 ∘ 2nd )) = frecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, (⦋𝐴 / 𝑥⦌𝐹 ∘ 2nd )))
9	2, 3, 7, 8	mp3an12i 1468	. . 3 ⊢ (𝐴 ∈ 𝑉 → frecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌(𝐹 ∘ 2nd )) = frecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, (⦋𝐴 / 𝑥⦌𝐹 ∘ 2nd )))
10	1, 9	eqtrd 2771	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌frecs(𝑅, 𝐷, (𝐹 ∘ 2nd )) = frecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, (⦋𝐴 / 𝑥⦌𝐹 ∘ 2nd )))
11		df-wrecs 8262	. . 3 ⊢ wrecs(𝑅, 𝐷, 𝐹) = frecs(𝑅, 𝐷, (𝐹 ∘ 2nd ))
12	11	csbeq2i 3845	. 2 ⊢ ⦋𝐴 / 𝑥⦌wrecs(𝑅, 𝐷, 𝐹) = ⦋𝐴 / 𝑥⦌frecs(𝑅, 𝐷, (𝐹 ∘ 2nd ))
13		df-wrecs 8262	. 2 ⊢ wrecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝐹) = frecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, (⦋𝐴 / 𝑥⦌𝐹 ∘ 2nd ))
14	10, 12, 13	3eqtr4g 2796	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌wrecs(𝑅, 𝐷, 𝐹) = wrecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝐹))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ⦋csb 3837   ∘ ccom 5635  2^nd c2nd 7941  frecscfrecs 8230  wrecscwrecs 8261

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-xp 5637  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-iota 6454  df-fv 6506  df-ov 7370  df-frecs 8231  df-wrecs 8262

This theorem is referenced by:  csbrecsg  37644