Theorem reldmprcof1 50661

Description: The domain of the object part of the pre-composition functor is a relation. (Contributed by Zhi Wang, 2-Nov-2025.)

Assertion

Ref	Expression
reldmprcof1	⊢ Rel dom (1st ‘(𝑃 −∘F 𝐹))

Proof of Theorem reldmprcof1

Dummy variables 𝑎 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relfunc 18052	. . . 4 ⊢ Rel ((1st ‘𝑃) Func (2nd ‘𝑃))
2		ovex 7463	. . . . . 6 ⊢ (𝑘 ∘func 𝐹) ∈ V
3		eqid 2763	. . . . . 6 ⊢ (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)) = (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹))
4	2, 3	dmmpti 6694	. . . . 5 ⊢ dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)) = ((1st ‘𝑃) Func (2nd ‘𝑃))
5	4	releqi 5759	. . . 4 ⊢ (Rel dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)) ↔ Rel ((1st ‘𝑃) Func (2nd ‘𝑃)))
6	1, 5	mpbir 234	. . 3 ⊢ Rel dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹))
7		eqid 2763	. . . . . . 7 ⊢ ((1st ‘𝑃) Func (2nd ‘𝑃)) = ((1st ‘𝑃) Func (2nd ‘𝑃))
8		eqid 2763	. . . . . . 7 ⊢ ((1st ‘𝑃) Nat (2nd ‘𝑃)) = ((1st ‘𝑃) Nat (2nd ‘𝑃))
9		simpr 490	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → 𝐹 ∈ V)
10		simpl 488	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → 𝑃 ∈ V)
11		eqidd 2764	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (1st ‘𝑃) = (1st ‘𝑃))
12		eqidd 2764	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (2nd ‘𝑃) = (2nd ‘𝑃))
13	7, 8, 9, 10, 11, 12	prcofvalg 50656	. . . . . 6 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (𝑃 −∘F 𝐹) = ⟨(𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑙 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑎 ∈ (𝑘((1st ‘𝑃) Nat (2nd ‘𝑃))𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)
14		ovex 7463	. . . . . . . 8 ⊢ ((1st ‘𝑃) Func (2nd ‘𝑃)) ∈ V
15	14	mptex 7239	. . . . . . 7 ⊢ (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)) ∈ V
16	14, 14	mpoex 8103	. . . . . . 7 ⊢ (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑙 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑎 ∈ (𝑘((1st ‘𝑃) Nat (2nd ‘𝑃))𝑙) ↦ (𝑎 ∘ (1st ‘𝐹)))) ∈ V
17	15, 16	op1std 8021	. . . . . 6 ⊢ ((𝑃 −∘F 𝐹) = ⟨(𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑙 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑎 ∈ (𝑘((1st ‘𝑃) Nat (2nd ‘𝑃))𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩ → (1st ‘(𝑃 −∘F 𝐹)) = (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)))
18	13, 17	syl 18	. . . . 5 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (1st ‘(𝑃 −∘F 𝐹)) = (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)))
19	18	dmeqd 5892	. . . 4 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (1st ‘(𝑃 −∘F 𝐹)) = dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)))
20	19	releqd 5760	. . 3 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (Rel dom (1st ‘(𝑃 −∘F 𝐹)) ↔ Rel dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹))))
21	6, 20	mpbiri 261	. 2 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → Rel dom (1st ‘(𝑃 −∘F 𝐹)))
22		rel0 5781	. . 3 ⊢ Rel ∅
23		reldmprcof 50655	. . . . . . . . 9 ⊢ Rel dom −∘F
24	23	ovprc 7468	. . . . . . . 8 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (𝑃 −∘F 𝐹) = ∅)
25	24	fveq2d 6900	. . . . . . 7 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (1st ‘(𝑃 −∘F 𝐹)) = (1st ‘∅))
26		1st0 8017	. . . . . . 7 ⊢ (1st ‘∅) = ∅
27	25, 26	eqtrdi 2814	. . . . . 6 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (1st ‘(𝑃 −∘F 𝐹)) = ∅)
28	27	dmeqd 5892	. . . . 5 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (1st ‘(𝑃 −∘F 𝐹)) = dom ∅)
29		dm0 5907	. . . . 5 ⊢ dom ∅ = ∅
30	28, 29	eqtrdi 2814	. . . 4 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (1st ‘(𝑃 −∘F 𝐹)) = ∅)
31	30	releqd 5760	. . 3 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (Rel dom (1st ‘(𝑃 −∘F 𝐹)) ↔ Rel ∅))
32	22, 31	mpbiri 261	. 2 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → Rel dom (1st ‘(𝑃 −∘F 𝐹)))
33	21, 32	pm2.61i 184	1 ⊢ Rel dom (1st ‘(𝑃 −∘F 𝐹))

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1569   ∈ wcel 2146  Vcvv 3454  ∅c0 4281  ⟨cop 4838   ↦ cmpt 5189  dom cdm 5655   ∘ ccom 5659  Rel wrel 5660  ‘cfv 6547  (class class class)co 7430   ∈ cmpo 7432  1^st c1st 8009  2^nd c2nd 8010   Func cfunc 18044   ∘_func ccofu 18046   Nat cnat 18134   −∘_F cprcof 50653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8011  df-2nd 8012  df-func 18048  df-prcof 50654

This theorem is used by: (None)