Theorem evl1fval1lem 20493

Description: Lemma for evl1fval1 20494. (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 2821	. . 3 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
2		eqid 2821	. . 3 ⊢ (1o eval 𝑅) = (1o eval 𝑅)
3		evl1fval1.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
4	1, 2, 3	evl1fval 20491	. 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 6684	. . . . 5 ⊢ 𝐵 ∈ V
8	7	pwid 4563	. . . 4 ⊢ 𝐵 ∈ 𝒫 𝐵
9		eqid 2821	. . . . 5 ⊢ (𝑅 evalSub1 𝐵) = (𝑅 evalSub1 𝐵)
10		eqid 2821	. . . . 5 ⊢ (1o evalSub 𝑅) = (1o evalSub 𝑅)
11	9, 10, 3	evls1fval 20482	. . . 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 689	. . 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 20308	. . . . 5 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵)
14	13	eqcomi 2830	. . . 4 ⊢ ((1o evalSub 𝑅)‘𝐵) = (1o eval 𝑅)
15	14	coeq2i 5731	. . 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	syl6eq 2872	. 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 2882	1 ⊢ (𝑅 ∈ 𝑉 → 𝑄 = (𝑅 evalSub1 𝐵))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  𝒫 cpw 4539  {csn 4567   ↦ cmpt 5146   × cxp 5553   ∘ ccom 5559  ‘cfv 6355  (class class class)co 7156  1_oc1o 8095   ↑_m cmap 8406  Basecbs 16483   evalSub ces 20284   eval cevl 20285   evalSub₁ ces1 20476  eval₁ce1 20477

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-evls 20286  df-evl 20287  df-evls1 20478  df-evl1 20479

This theorem is referenced by:  evl1fval1  20494