Theorem reldmprcof 50212

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 50211	. 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 7553	1 ⊢ Rel dom −∘F

This proof depends on syntax axioms:  Vcvv 3457  ⦋csb 3854  ⟨cop 4597   ↦ cmpt 5194  dom cdm 5663   ∘ ccom 5667  Rel wrel 5668  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  1^st c1st 7990  2^nd c2nd 7991   Func cfunc 17935   ∘_func ccofu 17937   Nat cnat 18025   −∘_F cprcof 50210

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-dm 5673  df-oprab 7423  df-mpo 7424  df-prcof 50211

This theorem is used by:  reldmprcof1  50218  reldmprcof2  50219  prcof1  50225