Theorem reldmprcof 49850

Description: The domain of −∘_F is a relation. (Contributed by Zhi Wang, 2-Nov-2025.)

Assertion

Ref	Expression
reldmprcof	⊢ Rel dom −∘F

Proof of Theorem reldmprcof

Dummy variables 𝑎 𝑏 𝑑 𝑒 𝑓 𝑘 𝑙 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-prcof 49849	. 2 ⊢ −∘F = (𝑝 ∈ V, 𝑓 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑑⦌⦋(2nd ‘𝑝) / 𝑒⦌⦋(𝑑 Func 𝑒) / 𝑏⦌⟨(𝑘 ∈ 𝑏 ↦ (𝑘 ∘func 𝑓)), (𝑘 ∈ 𝑏, 𝑙 ∈ 𝑏 ↦ (𝑎 ∈ (𝑘(𝑑 Nat 𝑒)𝑙) ↦ (𝑎 ∘ (1st ‘𝑓))))⟩)
2	1	reldmmpo 7501	1 ⊢ Rel dom −∘F

Syntax hints:  Vcvv 3429  ⦋csb 3837  ⟨cop 4573   ↦ cmpt 5166  dom cdm 5631   ∘ ccom 5635  Rel wrel 5636  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  1^st c1st 7940  2^nd c2nd 7941   Func cfunc 17821   ∘_func ccofu 17823   Nat cnat 17911   −∘_F cprcof 49848

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-dm 5641  df-oprab 7371  df-mpo 7372  df-prcof 49849

This theorem is referenced by:  reldmprcof1  49856  reldmprcof2  49857  prcof1  49863