Theorem evl1fval1lem 22215

Description: Lemma for evl1fval1 22216. (Contributed by AV, 11-Sep-2019.)

Hypotheses

Ref	Expression
evl1fval1.q	⊢ 𝑄 = (eval1‘𝑅)
evl1fval1.b	⊢ 𝐵 = (Base‘𝑅)

Assertion

Ref	Expression
evl1fval1lem	⊢ (𝑅 ∈ 𝑉 → 𝑄 = (𝑅 evalSub1 𝐵))

Proof of Theorem evl1fval1lem

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . 3 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
2		eqid 2729	. . 3 ⊢ (1o eval 𝑅) = (1o eval 𝑅)
3		evl1fval1.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
4	1, 2, 3	evl1fval 22213	. 2 ⊢ (eval1‘𝑅) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅))
5		evl1fval1.q	. . 3 ⊢ 𝑄 = (eval1‘𝑅)
6	5	a1i 11	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑄 = (eval1‘𝑅))
7	3	fvexi 6836	. . . . 5 ⊢ 𝐵 ∈ V
8	7	pwid 4573	. . . 4 ⊢ 𝐵 ∈ 𝒫 𝐵
9		eqid 2729	. . . . 5 ⊢ (𝑅 evalSub1 𝐵) = (𝑅 evalSub1 𝐵)
10		eqid 2729	. . . . 5 ⊢ (1o evalSub 𝑅) = (1o evalSub 𝑅)
11	9, 10, 3	evls1fval 22204	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐵 ∈ 𝒫 𝐵) → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑅)‘𝐵)))
12	8, 11	mpan2 691	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑅)‘𝐵)))
13	2, 3	evlval 22000	. . . . 5 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵)
14	13	eqcomi 2738	. . . 4 ⊢ ((1o evalSub 𝑅)‘𝐵) = (1o eval 𝑅)
15	14	coeq2i 5803	. . 3 ⊢ ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑅)‘𝐵)) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅))
16	12, 15	eqtrdi 2780	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅)))
17	4, 6, 16	3eqtr4a 2790	1 ⊢ (𝑅 ∈ 𝑉 → 𝑄 = (𝑅 evalSub1 𝐵))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  𝒫 cpw 4551  {csn 4577   ↦ cmpt 5173   × cxp 5617   ∘ ccom 5623  ‘cfv 6482  (class class class)co 7349  1_oc1o 8381   ↑_m cmap 8753  Basecbs 17120   evalSub ces 21977   eval cevl 21978   evalSub₁ ces1 22198  eval₁ce1 22199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-evls 21979  df-evl 21980  df-evls1 22200  df-evl1 22201

This theorem is referenced by:  evl1fval1  22216