reldmprcof2 - Mathbox for Zhi Wang

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Theorem reldmprcof2 49872

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 2739	. . . 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 7490	. . 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 2739	. . . . . . 7 ⊢ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) = ((1^st ‘𝑃) Func (2^nd ‘𝑃))
4		eqid 2739	. . . . . . 7 ⊢ ((1^st ‘𝑃) Nat (2^nd ‘𝑃)) = ((1^st ‘𝑃) Nat (2^nd ‘𝑃))
5		simpr 485	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → 𝐹 ∈ V)
6		simpl 483	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → 𝑃 ∈ V)
7		eqidd 2740	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (1^st ‘𝑃) = (1^st ‘𝑃))
8		eqidd 2740	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (2^nd ‘𝑃) = (2^nd ‘𝑃))
9	3, 4, 5, 6, 7, 8	prcofvalg 49866	. . . . . 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 7389	. . . . . . . 8 ⊢ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ∈ V
11	10	mptex 7167	. . . . . . 7 ⊢ (𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑘 ∘_func 𝐹)) ∈ V
12	10, 10	mpoex 8021	. . . . . . 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 7942	. . . . . 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 5847	. . . 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 5722	. . 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 259	. 2 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → Rel dom (2^nd ‘(𝑃 −∘_F 𝐹)))
18		rel0 5742	. . 3 ⊢ Rel ∅
19		reldmprcof 49865	. . . . . . . . 9 ⊢ Rel dom −∘_F
20	19	ovprc 7394	. . . . . . . 8 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (𝑃 −∘_F 𝐹) = ∅)
21	20	fveq2d 6831	. . . . . . 7 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (2^nd ‘(𝑃 −∘_F 𝐹)) = (2^nd ‘∅))
22		2nd0 7938	. . . . . . 7 ⊢ (2^nd ‘∅) = ∅
23	21, 22	eqtrdi 2790	. . . . . 6 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (2^nd ‘(𝑃 −∘_F 𝐹)) = ∅)
24	23	dmeqd 5847	. . . . 5 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (2^nd ‘(𝑃 −∘_F 𝐹)) = dom ∅)
25		dm0 5862	. . . . 5 ⊢ dom ∅ = ∅
26	24, 25	eqtrdi 2790	. . . 4 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (2^nd ‘(𝑃 −∘_F 𝐹)) = ∅)
27	26	releqd 5722	. . 3 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (Rel dom (2^nd ‘(𝑃 −∘_F 𝐹)) ↔ Rel ∅))
28	18, 27	mpbiri 259	. 2 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → Rel dom (2^nd ‘(𝑃 −∘_F 𝐹)))
29	17, 28	pm2.61i 183	1 ⊢ Rel dom (2^nd ‘(𝑃 −∘_F 𝐹))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1547 ∈ wcel 2119 Vcvv 3431 ∅c0 4261 ⟨cop 4561 ↦ cmpt 5153 dom cdm 5618 ∘ ccom 5622 Rel wrel 5623 ‘cfv 6485 (class class class)co 7356 ∈ cmpo 7358 1^st c1st 7929 2^nd c2nd 7930 Func cfunc 17812 ∘_func ccofu 17814 Nat cnat 17902 −∘_F cprcof 49863

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 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678

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 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7931 df-2nd 7932 df-prcof 49864

This theorem is referenced by: (None)

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