Theorem reldmprcof 49734

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 49733	. 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 7502	1 ⊢ Rel dom −∘F

Syntax hints:  Vcvv 3442  ⦋csb 3851  ⟨cop 4588   ↦ cmpt 5181  dom cdm 5632   ∘ ccom 5636  Rel wrel 5637  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  1^st c1st 7941  2^nd c2nd 7942   Func cfunc 17790   ∘_func ccofu 17792   Nat cnat 17880   −∘_F cprcof 49732

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 2709  ax-sep 5243  ax-pr 5379

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-dm 5642  df-oprab 7372  df-mpo 7373  df-prcof 49733

This theorem is referenced by:  reldmprcof1  49740  reldmprcof2  49741  prcof1  49747