Theorem relfldext 33943

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 33940	. 2 ⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
2	1	relopabiv 5794	1 ⊢ Rel /FldExt

Syntax hints:   ∧ wa 399   = wceq 1561   ∈ wcel 2143  Rel wrel 5653  ‘cfv 6521  (class class class)co 7396  Basecbs 17255   ↾_s cress 17276  SubRingcsubrg 20629  Fieldcfield 20789  /_FldExtcfldext 33937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1564  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-ss 3922  df-opab 5164  df-xp 5654  df-rel 5655  df-fldext 33940

This theorem is referenced by:  extdgval  33952