Theorem reldmprcof1 50179

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 17914	. . . 4 ⊢ Rel ((1st ‘𝑃) Func (2nd ‘𝑃))
2		ovex 7443	. . . . . 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 6679	. . . . 5 ⊢ dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)) = ((1st ‘𝑃) Func (2nd ‘𝑃))
5	4	releqi 5764	. . . 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 489	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → 𝐹 ∈ V)
10		simpl 487	. . . . . . 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 50174	. . . . . 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 7443	. . . . . . . 8 ⊢ ((1st ‘𝑃) Func (2nd ‘𝑃)) ∈ V
15	14	mptex 7221	. . . . . . 7 ⊢ (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)) ∈ V
16	14, 14	mpoex 8072	. . . . . . 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 7992	. . . . . 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 5895	. . . 4 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (1st ‘(𝑃 −∘F 𝐹)) = dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)))
20	19	releqd 5765	. . 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 5785	. . 3 ⊢ Rel ∅
23		reldmprcof 50173	. . . . . . . . 9 ⊢ Rel dom −∘F
24	23	ovprc 7448	. . . . . . . 8 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (𝑃 −∘F 𝐹) = ∅)
25	24	fveq2d 6885	. . . . . . 7 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (1st ‘(𝑃 −∘F 𝐹)) = (1st ‘∅))
26		1st0 7988	. . . . . . 7 ⊢ (1st ‘∅) = ∅
27	25, 26	eqtrdi 2814	. . . . . 6 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (1st ‘(𝑃 −∘F 𝐹)) = ∅)
28	27	dmeqd 5895	. . . . 5 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (1st ‘(𝑃 −∘F 𝐹)) = dom ∅)
29		dm0 5910	. . . . 5 ⊢ dom ∅ = ∅
30	28, 29	eqtrdi 2814	. . . 4 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (1st ‘(𝑃 −∘F 𝐹)) = ∅)
31	30	releqd 5765	. . 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))

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ∅c0 4286  ⟨cop 4595   ↦ cmpt 5192  dom cdm 5661   ∘ ccom 5665  Rel wrel 5666  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7980  2^nd c2nd 7981   Func cfunc 17906   ∘_func ccofu 17908   Nat cnat 17996   −∘_F cprcof 50171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-func 17910  df-prcof 50172

This theorem is referenced by: (None)