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Theorem evl1fval1lem 20995
 Description: Lemma for evl1fval1 20996. (Contributed by AV, 11-Sep-2019.)
Hypotheses
Ref Expression
evl1fval1.q 𝑄 = (eval1𝑅)
evl1fval1.b 𝐵 = (Base‘𝑅)
Assertion
Ref Expression
evl1fval1lem (𝑅𝑉𝑄 = (𝑅 evalSub1 𝐵))

Proof of Theorem evl1fval1lem
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2798 . . 3 (eval1𝑅) = (eval1𝑅)
2 eqid 2798 . . 3 (1o eval 𝑅) = (1o eval 𝑅)
3 evl1fval1.b . . 3 𝐵 = (Base‘𝑅)
41, 2, 3evl1fval 20993 . 2 (eval1𝑅) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅))
5 evl1fval1.q . . 3 𝑄 = (eval1𝑅)
65a1i 11 . 2 (𝑅𝑉𝑄 = (eval1𝑅))
73fvexi 6669 . . . . 5 𝐵 ∈ V
87pwid 4524 . . . 4 𝐵 ∈ 𝒫 𝐵
9 eqid 2798 . . . . 5 (𝑅 evalSub1 𝐵) = (𝑅 evalSub1 𝐵)
10 eqid 2798 . . . . 5 (1o evalSub 𝑅) = (1o evalSub 𝑅)
119, 10, 3evls1fval 20984 . . . 4 ((𝑅𝑉𝐵 ∈ 𝒫 𝐵) → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑅)‘𝐵)))
128, 11mpan2 690 . . 3 (𝑅𝑉 → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑅)‘𝐵)))
132, 3evlval 20804 . . . . 5 (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵)
1413eqcomi 2807 . . . 4 ((1o evalSub 𝑅)‘𝐵) = (1o eval 𝑅)
1514coeq2i 5699 . . 3 ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑅)‘𝐵)) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅))
1612, 15eqtrdi 2849 . 2 (𝑅𝑉 → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅)))
174, 6, 163eqtr4a 2859 1 (𝑅𝑉𝑄 = (𝑅 evalSub1 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  𝒫 cpw 4500  {csn 4528   ↦ cmpt 5114   × cxp 5521   ∘ ccom 5527  ‘cfv 6332  (class class class)co 7145  1oc1o 8096   ↑m cmap 8407  Basecbs 16495   evalSub ces 20779   eval cevl 20780   evalSub1 ces1 20978  eval1ce1 20979 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-evls 20781  df-evl 20782  df-evls1 20980  df-evl1 20981 This theorem is referenced by:  evl1fval1  20996
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