Theorem reldmoppf 49931

Description: The domain of oppFunc is a relation. (Contributed by Zhi Wang, 13-Nov-2025.)

Assertion

Ref	Expression
reldmoppf	⊢ Rel dom oppFunc

Proof of Theorem reldmoppf

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-oppf 49929	. 2 ⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
2	1	reldmmpo 7546	1 ⊢ Rel dom oppFunc

This proof depends on syntax axioms:   ∧ wa 400  Vcvv 3454  ∅c0 4285  ifcif 4486  ⟨cop 4594  dom cdm 5660  Rel wrel 5665  tpos ctpos 8219   oppFunc coppf 49928

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-dm 5670  df-oprab 7416  df-mpo 7417  df-oppf 49929

This theorem is used by: (None)