Theorem evl1fval1lem 22703

Description: Lemma for evl1fval1 22704. (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 2763	. . 3 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
2		eqid 2763	. . 3 ⊢ (1o eval 𝑅) = (1o eval 𝑅)
3		evl1fval1.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
4	1, 2, 3	evl1fval 22701	. 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 6930	. . . . 5 ⊢ 𝐵 ∈ V
8	7	pwid 4584	. . . 4 ⊢ 𝐵 ∈ 𝒫 𝐵
9		eqid 2763	. . . . 5 ⊢ (𝑅 evalSub1 𝐵) = (𝑅 evalSub1 𝐵)
10		eqid 2763	. . . . 5 ⊢ (1o evalSub 𝑅) = (1o evalSub 𝑅)
11	9, 10, 3	evls1fval 22692	. . . 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 704	. . 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 22464	. . . . 5 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵)
14	13	eqcomi 2772	. . . 4 ⊢ ((1o evalSub 𝑅)‘𝐵) = (1o eval 𝑅)
15	14	coeq2i 5849	. . 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 2814	. 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 2824	1 ⊢ (𝑅 ∈ 𝑉 → 𝑄 = (𝑅 evalSub1 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  𝒫 cpw 4560  {csn 4588   ↦ cmpt 5192   × cxp 5656   ∘ ccom 5662  ‘cfv 6566  (class class class)co 7451  1_oc1o 8507   ↑_m cmap 8885  Basecbs 17431   evalSub ces 22436   eval cevl 22437   evalSub₁ ces1 22686  eval₁ce1 22687

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-evls 22438  df-evl 22439  df-evls1 22688  df-evl1 22689

This theorem is used by:  evl1fval1  22704