Theorem reldmevls1 22232

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 22230	. 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 7480	1 ⊢ Rel dom evalSub1

Syntax hints:  Vcvv 3436  ⦋csb 3845  𝒫 cpw 4547  {csn 4573   ↦ cmpt 5170   × cxp 5612  dom cdm 5614   ∘ ccom 5618  Rel wrel 5619  ‘cfv 6481  (class class class)co 7346  1_oc1o 8378   ↑_m cmap 8750  Basecbs 17120   evalSub ces 22007   evalSub₁ ces1 22228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-dm 5624  df-oprab 7350  df-mpo 7351  df-evls1 22230

This theorem is referenced by:  evl1fval1  22246