Theorem reldmprcof1 49568

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 17784	. . . 4 ⊢ Rel ((1st ‘𝑃) Func (2nd ‘𝑃))
2		ovex 7389	. . . . . 6 ⊢ (𝑘 ∘func 𝐹) ∈ V
3		eqid 2734	. . . . . 6 ⊢ (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)) = (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹))
4	2, 3	dmmpti 6634	. . . . 5 ⊢ dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)) = ((1st ‘𝑃) Func (2nd ‘𝑃))
5	4	releqi 5725	. . . 4 ⊢ (Rel dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)) ↔ Rel ((1st ‘𝑃) Func (2nd ‘𝑃)))
6	1, 5	mpbir 231	. . 3 ⊢ Rel dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹))
7		eqid 2734	. . . . . . 7 ⊢ ((1st ‘𝑃) Func (2nd ‘𝑃)) = ((1st ‘𝑃) Func (2nd ‘𝑃))
8		eqid 2734	. . . . . . 7 ⊢ ((1st ‘𝑃) Nat (2nd ‘𝑃)) = ((1st ‘𝑃) Nat (2nd ‘𝑃))
9		simpr 484	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → 𝐹 ∈ V)
10		simpl 482	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → 𝑃 ∈ V)
11		eqidd 2735	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (1st ‘𝑃) = (1st ‘𝑃))
12		eqidd 2735	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (2nd ‘𝑃) = (2nd ‘𝑃))
13	7, 8, 9, 10, 11, 12	prcofvalg 49563	. . . . . 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 7389	. . . . . . . 8 ⊢ ((1st ‘𝑃) Func (2nd ‘𝑃)) ∈ V
15	14	mptex 7167	. . . . . . 7 ⊢ (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)) ∈ V
16	14, 14	mpoex 8021	. . . . . . 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 7941	. . . . . 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 5852	. . . 4 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (1st ‘(𝑃 −∘F 𝐹)) = dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)))
20	19	releqd 5726	. . 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 258	. 2 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → Rel dom (1st ‘(𝑃 −∘F 𝐹)))
22		rel0 5746	. . 3 ⊢ Rel ∅
23		reldmprcof 49562	. . . . . . . . 9 ⊢ Rel dom −∘F
24	23	ovprc 7394	. . . . . . . 8 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (𝑃 −∘F 𝐹) = ∅)
25	24	fveq2d 6836	. . . . . . 7 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (1st ‘(𝑃 −∘F 𝐹)) = (1st ‘∅))
26		1st0 7937	. . . . . . 7 ⊢ (1st ‘∅) = ∅
27	25, 26	eqtrdi 2785	. . . . . 6 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (1st ‘(𝑃 −∘F 𝐹)) = ∅)
28	27	dmeqd 5852	. . . . 5 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (1st ‘(𝑃 −∘F 𝐹)) = dom ∅)
29		dm0 5867	. . . . 5 ⊢ dom ∅ = ∅
30	28, 29	eqtrdi 2785	. . . 4 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (1st ‘(𝑃 −∘F 𝐹)) = ∅)
31	30	releqd 5726	. . 3 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (Rel dom (1st ‘(𝑃 −∘F 𝐹)) ↔ Rel ∅))
32	22, 31	mpbiri 258	. 2 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → Rel dom (1st ‘(𝑃 −∘F 𝐹)))
33	21, 32	pm2.61i 182	1 ⊢ Rel dom (1st ‘(𝑃 −∘F 𝐹))

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3438  ∅c0 4283  ⟨cop 4584   ↦ cmpt 5177  dom cdm 5622   ∘ ccom 5626  Rel wrel 5627  ‘cfv 6490  (class class class)co 7356   ∈ cmpo 7358  1^st c1st 7929  2^nd c2nd 7930   Func cfunc 17776   ∘_func ccofu 17778   Nat cnat 17866   −∘_F cprcof 49560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-func 17780  df-prcof 49561

This theorem is referenced by: (None)