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Theorem evl1fval1lem 19515
 Description: Lemma for evl1fval1 19516. (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 2610 . . 3 (eval1𝑅) = (eval1𝑅)
2 eqid 2610 . . 3 (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅)
3 evl1fval1.b . . 3 𝐵 = (Base‘𝑅)
41, 2, 3evl1fval 19513 . 2 (eval1𝑅) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑅))
5 evl1fval1.q . . 3 𝑄 = (eval1𝑅)
65a1i 11 . 2 (𝑅𝑉𝑄 = (eval1𝑅))
7 fvex 6113 . . . . . 6 (Base‘𝑅) ∈ V
83, 7eqeltri 2684 . . . . 5 𝐵 ∈ V
98pwid 4122 . . . 4 𝐵 ∈ 𝒫 𝐵
10 eqid 2610 . . . . 5 (𝑅 evalSub1 𝐵) = (𝑅 evalSub1 𝐵)
11 eqid 2610 . . . . 5 (1𝑜 evalSub 𝑅) = (1𝑜 evalSub 𝑅)
1210, 11, 3evls1fval 19505 . . . 4 ((𝑅𝑉𝐵 ∈ 𝒫 𝐵) → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑅)‘𝐵)))
139, 12mpan2 703 . . 3 (𝑅𝑉 → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑅)‘𝐵)))
142, 3evlval 19345 . . . . 5 (1𝑜 eval 𝑅) = ((1𝑜 evalSub 𝑅)‘𝐵)
1514eqcomi 2619 . . . 4 ((1𝑜 evalSub 𝑅)‘𝐵) = (1𝑜 eval 𝑅)
1615coeq2i 5204 . . 3 ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑅)‘𝐵)) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑅))
1713, 16syl6eq 2660 . 2 (𝑅𝑉 → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑅)))
184, 6, 173eqtr4a 2670 1 (𝑅𝑉𝑄 = (𝑅 evalSub1 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  𝒫 cpw 4108  {csn 4125   ↦ cmpt 4643   × cxp 5036   ∘ ccom 5042  ‘cfv 5804  (class class class)co 6549  1𝑜c1o 7440   ↑𝑚 cmap 7744  Basecbs 15695   evalSub ces 19325   eval cevl 19326   evalSub1 ces1 19499  eval1ce1 19500 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-evls 19327  df-evl 19328  df-evls1 19501  df-evl1 19502 This theorem is referenced by:  evl1fval1  19516
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