Theorem relfldext 33647

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 33644	. 2 ⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
2	1	relopabiv 5758	1 ⊢ Rel /FldExt

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2110  Rel wrel 5619  ‘cfv 6477  (class class class)co 7341  Basecbs 17112   ↾_s cress 17133  SubRingcsubrg 20477  Fieldcfield 20638  /_FldExtcfldext 33641

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 2112  ax-9 2120  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3436  df-ss 3917  df-opab 5152  df-xp 5620  df-rel 5621  df-fldext 33644

This theorem is referenced by:  extdgval  33656