Theorem reldmprcof 49562

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 49561	. 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 7490	1 ⊢ Rel dom −∘F

Syntax hints:  Vcvv 3438  ⦋csb 3847  ⟨cop 4584   ↦ cmpt 5177  dom cdm 5622   ∘ ccom 5626  Rel wrel 5627  ‘cfv 6490  (class class class)co 7356   ∈ cmpo 7358  1^st c1st 7929  2^nd c2nd 7930   Func cfunc 17776   ∘_func ccofu 17778   Nat cnat 17866   −∘_F cprcof 49560

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-xp 5628  df-rel 5629  df-dm 5632  df-oprab 7360  df-mpo 7361  df-prcof 49561

This theorem is referenced by:  reldmprcof1  49568  reldmprcof2  49569  prcof1  49575