Theorem evl1fval1lem 22274

Description: Lemma for evl1fval1 22275. (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 2736	. . 3 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
2		eqid 2736	. . 3 ⊢ (1o eval 𝑅) = (1o eval 𝑅)
3		evl1fval1.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
4	1, 2, 3	evl1fval 22272	. 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 6848	. . . . 5 ⊢ 𝐵 ∈ V
8	7	pwid 4576	. . . 4 ⊢ 𝐵 ∈ 𝒫 𝐵
9		eqid 2736	. . . . 5 ⊢ (𝑅 evalSub1 𝐵) = (𝑅 evalSub1 𝐵)
10		eqid 2736	. . . . 5 ⊢ (1o evalSub 𝑅) = (1o evalSub 𝑅)
11	9, 10, 3	evls1fval 22263	. . . 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 22055	. . . . 5 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵)
14	13	eqcomi 2745	. . . 4 ⊢ ((1o evalSub 𝑅)‘𝐵) = (1o eval 𝑅)
15	14	coeq2i 5809	. . 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 2787	. 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 2797	1 ⊢ (𝑅 ∈ 𝑉 → 𝑄 = (𝑅 evalSub1 𝐵))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  𝒫 cpw 4554  {csn 4580   ↦ cmpt 5179   × cxp 5622   ∘ ccom 5628  ‘cfv 6492  (class class class)co 7358  1_oc1o 8390   ↑_m cmap 8763  Basecbs 17136   evalSub ces 22027   eval cevl 22028   evalSub₁ ces1 22257  eval₁ce1 22258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-evls 22029  df-evl 22030  df-evls1 22259  df-evl1 22260

This theorem is referenced by:  evl1fval1  22275