Theorem evl1fval1lem 22305

Description: Lemma for evl1fval1 22306. (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 2737	. . 3 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
2		eqid 2737	. . 3 ⊢ (1o eval 𝑅) = (1o eval 𝑅)
3		evl1fval1.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
4	1, 2, 3	evl1fval 22303	. 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 4564	. . . 4 ⊢ 𝐵 ∈ 𝒫 𝐵
9		eqid 2737	. . . . 5 ⊢ (𝑅 evalSub1 𝐵) = (𝑅 evalSub1 𝐵)
10		eqid 2737	. . . . 5 ⊢ (1o evalSub 𝑅) = (1o evalSub 𝑅)
11	9, 10, 3	evls1fval 22294	. . . 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 692	. . 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 22088	. . . . 5 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵)
14	13	eqcomi 2746	. . . 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 2788	. 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 2798	1 ⊢ (𝑅 ∈ 𝑉 → 𝑄 = (𝑅 evalSub1 𝐵))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  𝒫 cpw 4542  {csn 4568   ↦ cmpt 5167   × cxp 5622   ∘ ccom 5628  ‘cfv 6492  (class class class)co 7360  1_oc1o 8391   ↑_m cmap 8766  Basecbs 17170   evalSub ces 22060   eval cevl 22061   evalSub₁ ces1 22288  eval₁ce1 22289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  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 7363  df-oprab 7364  df-mpo 7365  df-evls 22062  df-evl 22063  df-evls1 22290  df-evl1 22291

This theorem is referenced by:  evl1fval1  22306