Theorem relfldext 33691

Description: The field extension is a relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)

Assertion

Ref	Expression
relfldext	⊢ Rel /FldExt

Proof of Theorem relfldext

Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fldext 33687	. 2 ⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
2	1	relopabiv 5804	1 ⊢ Rel /FldExt

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Rel wrel 5664  ‘cfv 6536  (class class class)co 7410  Basecbs 17233   ↾_s cress 17256  SubRingcsubrg 20534  Fieldcfield 20695  /_FldExtcfldext 33683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-ss 3948  df-opab 5187  df-xp 5665  df-rel 5666  df-fldext 33687

This theorem is referenced by:  extdgval  33700