Theorem relfldext 31124

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 31120	. 2 ⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
2	1	relopabiv 5657	1 ⊢ Rel /FldExt

Syntax hints:   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Rel wrel 5524  ‘cfv 6324  (class class class)co 7135  Basecbs 16475   ↾_s cress 16476  Fieldcfield 19496  SubRingcsubrg 19524  /_FldExtcfldext 31116

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-opab 5093  df-xp 5525  df-rel 5526  df-fldext 31120

This theorem is referenced by:  extdgval  31132