Theorem relfldext 34156

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 34153	. 2 ⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
2	1	relopabiv 5805	1 ⊢ Rel /FldExt

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Rel wrel 5664  ‘cfv 6537  (class class class)co 7416  Basecbs 17305   ↾_s cress 17326  SubRingcsubrg 20735  Fieldcfield 20895  /_FldExtcfldext 34150

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-ss 3919  df-opab 5172  df-xp 5665  df-rel 5666  df-fldext 34153

This theorem is used by:  extdgval  34165