Theorem csbwrecsg 8321

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 8287	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌frecs(𝑅, 𝐷, (𝐹 ∘ 2nd )) = frecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌(𝐹 ∘ 2nd )))
2		eqid 2765	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌𝑅 = ⦋𝐴 / 𝑥⦌𝑅
3		eqid 2765	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌𝐷 = ⦋𝐴 / 𝑥⦌𝐷
4		csbcog 6302	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹 ∘ 2nd ) = (⦋𝐴 / 𝑥⦌𝐹 ∘ ⦋𝐴 / 𝑥⦌2nd ))
5		csbconstg 3873	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌2nd = 2nd )
6	5	coeq2d 5850	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐹 ∘ ⦋𝐴 / 𝑥⦌2nd ) = (⦋𝐴 / 𝑥⦌𝐹 ∘ 2nd ))
7	4, 6	eqtrd 2800	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹 ∘ 2nd ) = (⦋𝐴 / 𝑥⦌𝐹 ∘ 2nd ))
8		frecseq123 8285	. . . 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 1494	. . 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 2800	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌frecs(𝑅, 𝐷, (𝐹 ∘ 2nd )) = frecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, (⦋𝐴 / 𝑥⦌𝐹 ∘ 2nd )))
11		df-wrecs 8315	. . 3 ⊢ wrecs(𝑅, 𝐷, 𝐹) = frecs(𝑅, 𝐷, (𝐹 ∘ 2nd ))
12	11	csbeq2i 3862	. 2 ⊢ ⦋𝐴 / 𝑥⦌wrecs(𝑅, 𝐷, 𝐹) = ⦋𝐴 / 𝑥⦌frecs(𝑅, 𝐷, (𝐹 ∘ 2nd ))
13		df-wrecs 8315	. 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 2825	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌wrecs(𝑅, 𝐷, 𝐹) = wrecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝐹))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ⦋csb 3854   ∘ ccom 5667  2^nd c2nd 7991  frecscfrecs 8283  wrecscwrecs 8314

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-xp 5669  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-iota 6496  df-fv 6548  df-ov 7422  df-frecs 8284  df-wrecs 8315

This theorem is used by:  csbrecsg  38031