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Theorem evl1fval1lem 19613
Description: Lemma for evl1fval1 19614. (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 2621 . . 3 (eval1𝑅) = (eval1𝑅)
2 eqid 2621 . . 3 (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅)
3 evl1fval1.b . . 3 𝐵 = (Base‘𝑅)
41, 2, 3evl1fval 19611 . 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 6158 . . . . . 6 (Base‘𝑅) ∈ V
83, 7eqeltri 2694 . . . . 5 𝐵 ∈ V
98pwid 4145 . . . 4 𝐵 ∈ 𝒫 𝐵
10 eqid 2621 . . . . 5 (𝑅 evalSub1 𝐵) = (𝑅 evalSub1 𝐵)
11 eqid 2621 . . . . 5 (1𝑜 evalSub 𝑅) = (1𝑜 evalSub 𝑅)
1210, 11, 3evls1fval 19603 . . . 4 ((𝑅𝑉𝐵 ∈ 𝒫 𝐵) → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑅)‘𝐵)))
139, 12mpan2 706 . . 3 (𝑅𝑉 → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑅)‘𝐵)))
142, 3evlval 19443 . . . . 5 (1𝑜 eval 𝑅) = ((1𝑜 evalSub 𝑅)‘𝐵)
1514eqcomi 2630 . . . 4 ((1𝑜 evalSub 𝑅)‘𝐵) = (1𝑜 eval 𝑅)
1615coeq2i 5242 . . 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 2671 . 2 (𝑅𝑉 → (𝑅 evalSub1 𝐵) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑅)))
184, 6, 173eqtr4a 2681 1 (𝑅𝑉𝑄 = (𝑅 evalSub1 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3186  𝒫 cpw 4130  {csn 4148  cmpt 4673   × cxp 5072  ccom 5078  cfv 5847  (class class class)co 6604  1𝑜c1o 7498  𝑚 cmap 7802  Basecbs 15781   evalSub ces 19423   eval cevl 19424   evalSub1 ces1 19597  eval1ce1 19598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-evls 19425  df-evl 19426  df-evls1 19599  df-evl1 19600
This theorem is referenced by:  evl1fval1  19614
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