Theorem relfldext 31046

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 31042	. 2 ⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
2	1	relopabiv 5679	1 ⊢ Rel /FldExt

Syntax hints:   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Rel wrel 5546  ‘cfv 6341  (class class class)co 7142  Basecbs 16466   ↾_s cress 16467  Fieldcfield 19486  SubRingcsubrg 19514  /_FldExtcfldext 31038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3488  df-in 3931  df-ss 3940  df-opab 5115  df-xp 5547  df-rel 5548  df-fldext 31042

This theorem is referenced by:  extdgval  31054