Theorem csbwrecsg 8362

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 8325	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌frecs(𝑅, 𝐷, (𝐹 ∘ 2nd )) = frecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌(𝐹 ∘ 2nd )))
2		eqid 2740	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌𝑅 = ⦋𝐴 / 𝑥⦌𝑅
3		eqid 2740	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌𝐷 = ⦋𝐴 / 𝑥⦌𝐷
4		csbcog 6328	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹 ∘ 2nd ) = (⦋𝐴 / 𝑥⦌𝐹 ∘ ⦋𝐴 / 𝑥⦌2nd ))
5		csbconstg 3940	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌2nd = 2nd )
6	5	coeq2d 5887	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐹 ∘ ⦋𝐴 / 𝑥⦌2nd ) = (⦋𝐴 / 𝑥⦌𝐹 ∘ 2nd ))
7	4, 6	eqtrd 2780	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹 ∘ 2nd ) = (⦋𝐴 / 𝑥⦌𝐹 ∘ 2nd ))
8		frecseq123 8323	. . . 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 1465	. . 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 2780	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌frecs(𝑅, 𝐷, (𝐹 ∘ 2nd )) = frecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, (⦋𝐴 / 𝑥⦌𝐹 ∘ 2nd )))
11		df-wrecs 8353	. . 3 ⊢ wrecs(𝑅, 𝐷, 𝐹) = frecs(𝑅, 𝐷, (𝐹 ∘ 2nd ))
12	11	csbeq2i 3929	. 2 ⊢ ⦋𝐴 / 𝑥⦌wrecs(𝑅, 𝐷, 𝐹) = ⦋𝐴 / 𝑥⦌frecs(𝑅, 𝐷, (𝐹 ∘ 2nd ))
13		df-wrecs 8353	. 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 2805	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌wrecs(𝑅, 𝐷, 𝐹) = wrecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝐹))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ⦋csb 3921   ∘ ccom 5704  2^nd c2nd 8029  frecscfrecs 8321  wrecscwrecs 8352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-iota 6525  df-fv 6581  df-ov 7451  df-frecs 8322  df-wrecs 8353

This theorem is referenced by:  csbrecsg  37294