Theorem reldmevls1 22261

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 22259	. 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 7492	1 ⊢ Rel dom evalSub1

Syntax hints:  Vcvv 3440  ⦋csb 3849  𝒫 cpw 4554  {csn 4580   ↦ cmpt 5179   × cxp 5622  dom cdm 5624   ∘ ccom 5628  Rel wrel 5629  ‘cfv 6492  (class class class)co 7358  1_oc1o 8390   ↑_m cmap 8763  Basecbs 17136   evalSub ces 22027   evalSub₁ ces1 22257

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-dm 5634  df-oprab 7362  df-mpo 7363  df-evls1 22259

This theorem is referenced by:  evl1fval1  22275