reldmprcof2 - Mathbox for Zhi Wang

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Theorem reldmprcof2 49493

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

**Assertion**
Ref	Expression
reldmprcof2	⊢ Rel dom (2^nd ‘(𝑃 −∘_F 𝐹))

Proof of Theorem reldmprcof2 Dummy variables 𝑎 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2731	. . . 4 ⊢ (𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)), 𝑙 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑎 ∈ (𝑘((1^st ‘𝑃) Nat (2^nd ‘𝑃))𝑙) ↦ (𝑎 ∘ (1^st ‘𝐹)))) = (𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)), 𝑙 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑎 ∈ (𝑘((1^st ‘𝑃) Nat (2^nd ‘𝑃))𝑙) ↦ (𝑎 ∘ (1^st ‘𝐹))))
2	1	reldmmpo 7480	. . 3 ⊢ Rel dom (𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)), 𝑙 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑎 ∈ (𝑘((1^st ‘𝑃) Nat (2^nd ‘𝑃))𝑙) ↦ (𝑎 ∘ (1^st ‘𝐹))))
3		eqid 2731	. . . . . . 7 ⊢ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) = ((1^st ‘𝑃) Func (2^nd ‘𝑃))
4		eqid 2731	. . . . . . 7 ⊢ ((1^st ‘𝑃) Nat (2^nd ‘𝑃)) = ((1^st ‘𝑃) Nat (2^nd ‘𝑃))
5		simpr 484	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → 𝐹 ∈ V)
6		simpl 482	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → 𝑃 ∈ V)
7		eqidd 2732	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (1^st ‘𝑃) = (1^st ‘𝑃))
8		eqidd 2732	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (2^nd ‘𝑃) = (2^nd ‘𝑃))
9	3, 4, 5, 6, 7, 8	prcofvalg 49487	. . . . . 6 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (𝑃 −∘_F 𝐹) = ⟨(𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑘 ∘_func 𝐹)), (𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)), 𝑙 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑎 ∈ (𝑘((1^st ‘𝑃) Nat (2^nd ‘𝑃))𝑙) ↦ (𝑎 ∘ (1^st ‘𝐹))))⟩)
10		ovex 7379	. . . . . . . 8 ⊢ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ∈ V
11	10	mptex 7157	. . . . . . 7 ⊢ (𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑘 ∘_func 𝐹)) ∈ V
12	10, 10	mpoex 8011	. . . . . . 7 ⊢ (𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)), 𝑙 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑎 ∈ (𝑘((1^st ‘𝑃) Nat (2^nd ‘𝑃))𝑙) ↦ (𝑎 ∘ (1^st ‘𝐹)))) ∈ V
13	11, 12	op2ndd 7932	. . . . . 6 ⊢ ((𝑃 −∘_F 𝐹) = ⟨(𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑘 ∘_func 𝐹)), (𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)), 𝑙 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑎 ∈ (𝑘((1^st ‘𝑃) Nat (2^nd ‘𝑃))𝑙) ↦ (𝑎 ∘ (1^st ‘𝐹))))⟩ → (2^nd ‘(𝑃 −∘_F 𝐹)) = (𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)), 𝑙 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑎 ∈ (𝑘((1^st ‘𝑃) Nat (2^nd ‘𝑃))𝑙) ↦ (𝑎 ∘ (1^st ‘𝐹)))))
14	9, 13	syl 17	. . . . 5 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (2^nd ‘(𝑃 −∘_F 𝐹)) = (𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)), 𝑙 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑎 ∈ (𝑘((1^st ‘𝑃) Nat (2^nd ‘𝑃))𝑙) ↦ (𝑎 ∘ (1^st ‘𝐹)))))
15	14	dmeqd 5844	. . . 4 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (2^nd ‘(𝑃 −∘_F 𝐹)) = dom (𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)), 𝑙 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑎 ∈ (𝑘((1^st ‘𝑃) Nat (2^nd ‘𝑃))𝑙) ↦ (𝑎 ∘ (1^st ‘𝐹)))))
16	15	releqd 5718	. . 3 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (Rel dom (2^nd ‘(𝑃 −∘_F 𝐹)) ↔ Rel dom (𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)), 𝑙 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑎 ∈ (𝑘((1^st ‘𝑃) Nat (2^nd ‘𝑃))𝑙) ↦ (𝑎 ∘ (1^st ‘𝐹))))))
17	2, 16	mpbiri 258	. 2 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → Rel dom (2^nd ‘(𝑃 −∘_F 𝐹)))
18		rel0 5738	. . 3 ⊢ Rel ∅
19		reldmprcof 49486	. . . . . . . . 9 ⊢ Rel dom −∘_F
20	19	ovprc 7384	. . . . . . . 8 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (𝑃 −∘_F 𝐹) = ∅)
21	20	fveq2d 6826	. . . . . . 7 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (2^nd ‘(𝑃 −∘_F 𝐹)) = (2^nd ‘∅))
22		2nd0 7928	. . . . . . 7 ⊢ (2^nd ‘∅) = ∅
23	21, 22	eqtrdi 2782	. . . . . 6 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (2^nd ‘(𝑃 −∘_F 𝐹)) = ∅)
24	23	dmeqd 5844	. . . . 5 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (2^nd ‘(𝑃 −∘_F 𝐹)) = dom ∅)
25		dm0 5859	. . . . 5 ⊢ dom ∅ = ∅
26	24, 25	eqtrdi 2782	. . . 4 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (2^nd ‘(𝑃 −∘_F 𝐹)) = ∅)
27	26	releqd 5718	. . 3 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (Rel dom (2^nd ‘(𝑃 −∘_F 𝐹)) ↔ Rel ∅))
28	18, 27	mpbiri 258	. 2 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → Rel dom (2^nd ‘(𝑃 −∘_F 𝐹)))
29	17, 28	pm2.61i 182	1 ⊢ Rel dom (2^nd ‘(𝑃 −∘_F 𝐹))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1541 ∈ wcel 2111 Vcvv 3436 ∅c0 4280 ⟨cop 4579 ↦ cmpt 5170 dom cdm 5614 ∘ ccom 5618 Rel wrel 5619 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 1^st c1st 7919 2^nd c2nd 7920 Func cfunc 17761 ∘_func ccofu 17763 Nat cnat 17851 −∘_F cprcof 49484

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-prcof 49485

This theorem is referenced by: (None)

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