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Theorem evl1fval1lem 21848
Description: Lemma for evl1fval1 21849. (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 2732 . . 3 (eval1𝑅) = (eval1𝑅)
2 eqid 2732 . . 3 (1o eval 𝑅) = (1o eval 𝑅)
3 evl1fval1.b . . 3 𝐵 = (Base‘𝑅)
41, 2, 3evl1fval 21846 . 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 6905 . . . . 5 𝐵 ∈ V
87pwid 4624 . . . 4 𝐵 ∈ 𝒫 𝐵
9 eqid 2732 . . . . 5 (𝑅 evalSub1 𝐵) = (𝑅 evalSub1 𝐵)
10 eqid 2732 . . . . 5 (1o evalSub 𝑅) = (1o evalSub 𝑅)
119, 10, 3evls1fval 21837 . . . 4 ((𝑅𝑉𝐵 ∈ 𝒫 𝐵) → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑅)‘𝐵)))
128, 11mpan2 689 . . 3 (𝑅𝑉 → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑅)‘𝐵)))
132, 3evlval 21657 . . . . 5 (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵)
1413eqcomi 2741 . . . 4 ((1o evalSub 𝑅)‘𝐵) = (1o eval 𝑅)
1514coeq2i 5860 . . 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 2788 . 2 (𝑅𝑉 → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅)))
174, 6, 163eqtr4a 2798 1 (𝑅𝑉𝑄 = (𝑅 evalSub1 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  𝒫 cpw 4602  {csn 4628  cmpt 5231   × cxp 5674  ccom 5680  cfv 6543  (class class class)co 7408  1oc1o 8458  m cmap 8819  Basecbs 17143   evalSub ces 21632   eval cevl 21633   evalSub1 ces1 21831  eval1ce1 21832
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  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 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-evls 21634  df-evl 21635  df-evls1 21833  df-evl1 21834
This theorem is referenced by:  evl1fval1  21849
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