Theorem relfldext 33776

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 33773	. 2 ⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
2	1	relopabiv 5765	1 ⊢ Rel /FldExt

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Rel wrel 5625  ‘cfv 6487  (class class class)co 7356  Basecbs 17168   ↾_s cress 17189  SubRingcsubrg 20535  Fieldcfield 20696  /_FldExtcfldext 33770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-v 3429  df-ss 3902  df-opab 5137  df-xp 5626  df-rel 5627  df-fldext 33773

This theorem is referenced by:  extdgval  33785