Theorem reldmevls1 21835

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 21833	. 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 7542	1 ⊢ Rel dom evalSub1

Syntax hints:  Vcvv 3474  ⦋csb 3893  𝒫 cpw 4602  {csn 4628   ↦ cmpt 5231   × cxp 5674  dom cdm 5676   ∘ ccom 5680  Rel wrel 5681  ‘cfv 6543  (class class class)co 7408  1_oc1o 8458   ↑_m cmap 8819  Basecbs 17143   evalSub ces 21632   evalSub₁ ces1 21831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-dm 5686  df-oprab 7412  df-mpo 7413  df-evls1 21833

This theorem is referenced by:  evl1fval1  21849