Theorem csbwrecsg 8345

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 8308	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌frecs(𝑅, 𝐷, (𝐹 ∘ 2nd )) = frecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌(𝐹 ∘ 2nd )))
2		eqid 2735	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌𝑅 = ⦋𝐴 / 𝑥⦌𝑅
3		eqid 2735	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌𝐷 = ⦋𝐴 / 𝑥⦌𝐷
4		csbcog 6319	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹 ∘ 2nd ) = (⦋𝐴 / 𝑥⦌𝐹 ∘ ⦋𝐴 / 𝑥⦌2nd ))
5		csbconstg 3927	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌2nd = 2nd )
6	5	coeq2d 5876	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐹 ∘ ⦋𝐴 / 𝑥⦌2nd ) = (⦋𝐴 / 𝑥⦌𝐹 ∘ 2nd ))
7	4, 6	eqtrd 2775	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹 ∘ 2nd ) = (⦋𝐴 / 𝑥⦌𝐹 ∘ 2nd ))
8		frecseq123 8306	. . . 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 1464	. . 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 2775	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌frecs(𝑅, 𝐷, (𝐹 ∘ 2nd )) = frecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, (⦋𝐴 / 𝑥⦌𝐹 ∘ 2nd )))
11		df-wrecs 8336	. . 3 ⊢ wrecs(𝑅, 𝐷, 𝐹) = frecs(𝑅, 𝐷, (𝐹 ∘ 2nd ))
12	11	csbeq2i 3916	. 2 ⊢ ⦋𝐴 / 𝑥⦌wrecs(𝑅, 𝐷, 𝐹) = ⦋𝐴 / 𝑥⦌frecs(𝑅, 𝐷, (𝐹 ∘ 2nd ))
13		df-wrecs 8336	. 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 2800	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 2106  ⦋csb 3908   ∘ ccom 5693  2^nd c2nd 8012  frecscfrecs 8304  wrecscwrecs 8335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fv 6571  df-ov 7434  df-frecs 8305  df-wrecs 8336

This theorem is referenced by:  csbrecsg  37311