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Theorem evl1fval1lem 19916
Description: Lemma for evl1fval1 19917. (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 2760 . . 3 (eval1𝑅) = (eval1𝑅)
2 eqid 2760 . . 3 (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅)
3 evl1fval1.b . . 3 𝐵 = (Base‘𝑅)
41, 2, 3evl1fval 19914 . 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 6363 . . . . . 6 (Base‘𝑅) ∈ V
83, 7eqeltri 2835 . . . . 5 𝐵 ∈ V
98pwid 4318 . . . 4 𝐵 ∈ 𝒫 𝐵
10 eqid 2760 . . . . 5 (𝑅 evalSub1 𝐵) = (𝑅 evalSub1 𝐵)
11 eqid 2760 . . . . 5 (1𝑜 evalSub 𝑅) = (1𝑜 evalSub 𝑅)
1210, 11, 3evls1fval 19906 . . . 4 ((𝑅𝑉𝐵 ∈ 𝒫 𝐵) → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑅)‘𝐵)))
139, 12mpan2 709 . . 3 (𝑅𝑉 → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑅)‘𝐵)))
142, 3evlval 19746 . . . . 5 (1𝑜 eval 𝑅) = ((1𝑜 evalSub 𝑅)‘𝐵)
1514eqcomi 2769 . . . 4 ((1𝑜 evalSub 𝑅)‘𝐵) = (1𝑜 eval 𝑅)
1615coeq2i 5438 . . 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 2810 . 2 (𝑅𝑉 → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑅)))
184, 6, 173eqtr4a 2820 1 (𝑅𝑉𝑄 = (𝑅 evalSub1 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  𝒫 cpw 4302  {csn 4321  cmpt 4881   × cxp 5264  ccom 5270  cfv 6049  (class class class)co 6814  1𝑜c1o 7723  𝑚 cmap 8025  Basecbs 16079   evalSub ces 19726   eval cevl 19727   evalSub1 ces1 19900  eval1ce1 19901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-evls 19728  df-evl 19729  df-evls1 19902  df-evl1 19903
This theorem is referenced by:  evl1fval1  19917
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