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 7555	1 ⊢ Rel dom evalSub1

Syntax hints:  Vcvv 3461  ⦋csb 3889  𝒫 cpw 4604  {csn 4630   ↦ cmpt 5232   × cxp 5676  dom cdm 5678   ∘ ccom 5682  Rel wrel 5683  ‘cfv 6549  (class class class)co 7419  1_oc1o 8480   ↑_m cmap 8845  Basecbs 17183   evalSub ces 22038   evalSub₁ ces1 22257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-dm 5688  df-oprab 7423  df-mpo 7424  df-evls1 22259

This theorem is referenced by:  evl1fval1  22275