Theorem reldmevls1 22445

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.

Step	Hyp	Ref	Expression
1		df-evls1 22443	. 2 ⊢ evalSub1 = (𝑠 ∈ V, 𝑟 ∈ 𝒫 (Base‘𝑠) ↦ ⦋(Base‘𝑠) / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)))
2	1	reldmmpo 7545	1 ⊢ Rel dom evalSub1

Syntax hints:  Vcvv 3463  ⦋csb 3861  𝒫 cpw 4567  {csn 4594   ↦ cmpt 5196   × cxp 5660  dom cdm 5662   ∘ ccom 5666  Rel wrel 5667  ‘cfv 6537  (class class class)co 7411  1_oc1o 8445   ↑_m cmap 8823  Basecbs 17268   evalSub ces 22191   evalSub₁ ces1 22441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-dm 5672  df-oprab 7415  df-mpo 7416  df-evls1 22443

This theorem is referenced by:  evl1fval1  22459