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Theorem reldmevls1 19503
 Description: Well-behaved binary operation property of evalSub1. (Contributed by AV, 7-Sep-2019.)
Assertion
Ref Expression
reldmevls1 Rel dom evalSub1

Proof of Theorem reldmevls1
Dummy variables 𝑟 𝑏 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-evls1 19501 . 2 evalSub1 = (𝑠 ∈ V, 𝑟 ∈ 𝒫 (Base‘𝑠) ↦ (Base‘𝑠) / 𝑏((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑠)‘𝑟)))
21reldmmpt2 6669 1 Rel dom evalSub1
 Colors of variables: wff setvar class Syntax hints:  Vcvv 3173  ⦋csb 3499  𝒫 cpw 4108  {csn 4125   ↦ cmpt 4643   × cxp 5036  dom cdm 5038   ∘ ccom 5042  Rel wrel 5043  ‘cfv 5804  (class class class)co 6549  1𝑜c1o 7440   ↑𝑚 cmap 7744  Basecbs 15695   evalSub ces 19325   evalSub1 ces1 19499 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 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-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-dm 5048  df-oprab 6553  df-mpt2 6554  df-evls1 19501 This theorem is referenced by:  evl1fval1  19516
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