Theorem reldmprcof1 49879

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 17821	. . . 4 ⊢ Rel ((1st ‘𝑃) Func (2nd ‘𝑃))
2		ovex 7390	. . . . . 6 ⊢ (𝑘 ∘func 𝐹) ∈ V
3		eqid 2739	. . . . . 6 ⊢ (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)) = (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹))
4	2, 3	dmmpti 6630	. . . . 5 ⊢ dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)) = ((1st ‘𝑃) Func (2nd ‘𝑃))
5	4	releqi 5722	. . . 4 ⊢ (Rel dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)) ↔ Rel ((1st ‘𝑃) Func (2nd ‘𝑃)))
6	1, 5	mpbir 232	. . 3 ⊢ Rel dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹))
7		eqid 2739	. . . . . . 7 ⊢ ((1st ‘𝑃) Func (2nd ‘𝑃)) = ((1st ‘𝑃) Func (2nd ‘𝑃))
8		eqid 2739	. . . . . . 7 ⊢ ((1st ‘𝑃) Nat (2nd ‘𝑃)) = ((1st ‘𝑃) Nat (2nd ‘𝑃))
9		simpr 485	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → 𝐹 ∈ V)
10		simpl 483	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → 𝑃 ∈ V)
11		eqidd 2740	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (1st ‘𝑃) = (1st ‘𝑃))
12		eqidd 2740	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (2nd ‘𝑃) = (2nd ‘𝑃))
13	7, 8, 9, 10, 11, 12	prcofvalg 49874	. . . . . 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 7390	. . . . . . . 8 ⊢ ((1st ‘𝑃) Func (2nd ‘𝑃)) ∈ V
15	14	mptex 7168	. . . . . . 7 ⊢ (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)) ∈ V
16	14, 14	mpoex 8022	. . . . . . 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 7942	. . . . . 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 17	. . . . 5 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (1st ‘(𝑃 −∘F 𝐹)) = (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)))
19	18	dmeqd 5848	. . . 4 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (1st ‘(𝑃 −∘F 𝐹)) = dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)))
20	19	releqd 5723	. . 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 259	. 2 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → Rel dom (1st ‘(𝑃 −∘F 𝐹)))
22		rel0 5743	. . 3 ⊢ Rel ∅
23		reldmprcof 49873	. . . . . . . . 9 ⊢ Rel dom −∘F
24	23	ovprc 7395	. . . . . . . 8 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (𝑃 −∘F 𝐹) = ∅)
25	24	fveq2d 6832	. . . . . . 7 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (1st ‘(𝑃 −∘F 𝐹)) = (1st ‘∅))
26		1st0 7938	. . . . . . 7 ⊢ (1st ‘∅) = ∅
27	25, 26	eqtrdi 2790	. . . . . 6 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (1st ‘(𝑃 −∘F 𝐹)) = ∅)
28	27	dmeqd 5848	. . . . 5 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (1st ‘(𝑃 −∘F 𝐹)) = dom ∅)
29		dm0 5863	. . . . 5 ⊢ dom ∅ = ∅
30	28, 29	eqtrdi 2790	. . . 4 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (1st ‘(𝑃 −∘F 𝐹)) = ∅)
31	30	releqd 5723	. . 3 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (Rel dom (1st ‘(𝑃 −∘F 𝐹)) ↔ Rel ∅))
32	22, 31	mpbiri 259	. 2 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → Rel dom (1st ‘(𝑃 −∘F 𝐹)))
33	21, 32	pm2.61i 183	1 ⊢ Rel dom (1st ‘(𝑃 −∘F 𝐹))

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ∅c0 4262  ⟨cop 4562   ↦ cmpt 5154  dom cdm 5619   ∘ ccom 5623  Rel wrel 5624  ‘cfv 6486  (class class class)co 7357   ∈ cmpo 7359  1^st c1st 7930  2^nd c2nd 7931   Func cfunc 17813   ∘_func ccofu 17815   Nat cnat 17903   −∘_F cprcof 49871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7932  df-2nd 7933  df-func 17817  df-prcof 49872

This theorem is referenced by: (None)