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Theorem evl1fval1lem 21841
Description: Lemma for evl1fval1 21842. (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 2733 . . 3 (eval1𝑅) = (eval1𝑅)
2 eqid 2733 . . 3 (1o eval 𝑅) = (1o eval 𝑅)
3 evl1fval1.b . . 3 𝐵 = (Base‘𝑅)
41, 2, 3evl1fval 21839 . 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 6903 . . . . 5 𝐵 ∈ V
87pwid 4624 . . . 4 𝐵 ∈ 𝒫 𝐵
9 eqid 2733 . . . . 5 (𝑅 evalSub1 𝐵) = (𝑅 evalSub1 𝐵)
10 eqid 2733 . . . . 5 (1o evalSub 𝑅) = (1o evalSub 𝑅)
119, 10, 3evls1fval 21830 . . . 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 21650 . . . . 5 (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵)
1413eqcomi 2742 . . . 4 ((1o evalSub 𝑅)‘𝐵) = (1o eval 𝑅)
1514coeq2i 5859 . . 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 2789 . 2 (𝑅𝑉 → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅)))
174, 6, 163eqtr4a 2799 1 (𝑅𝑉𝑄 = (𝑅 evalSub1 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  𝒫 cpw 4602  {csn 4628  cmpt 5231   × cxp 5674  ccom 5680  cfv 6541  (class class class)co 7406  1oc1o 8456  m cmap 8817  Basecbs 17141   evalSub ces 21625   eval cevl 21626   evalSub1 ces1 21824  eval1ce1 21825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-evls 21627  df-evl 21628  df-evls1 21826  df-evl1 21827
This theorem is referenced by:  evl1fval1  21842
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