Theorem relfldext 33367

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 33363	. 2 ⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
2	1	relopabiv 5814	1 ⊢ Rel /FldExt

Syntax hints:   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Rel wrel 5675  ‘cfv 6541  (class class class)co 7414  Basecbs 17177   ↾_s cress 17206  SubRingcsubrg 20508  Fieldcfield 20627  /_FldExtcfldext 33359

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-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3465  df-in 3946  df-ss 3956  df-opab 5204  df-xp 5676  df-rel 5677  df-fldext 33363

This theorem is referenced by:  extdgval  33375