reldmprcof2 - Mathbox for Zhi Wang

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Theorem reldmprcof2 50003

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 2762	. . . 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 7530	. . 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 2762	. . . . . . 7 ⊢ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) = ((1^st ‘𝑃) Func (2^nd ‘𝑃))
4		eqid 2762	. . . . . . 7 ⊢ ((1^st ‘𝑃) Nat (2^nd ‘𝑃)) = ((1^st ‘𝑃) Nat (2^nd ‘𝑃))
5		simpr 488	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → 𝐹 ∈ V)
6		simpl 486	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → 𝑃 ∈ V)
7		eqidd 2763	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (1^st ‘𝑃) = (1^st ‘𝑃))
8		eqidd 2763	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (2^nd ‘𝑃) = (2^nd ‘𝑃))
9	3, 4, 5, 6, 7, 8	prcofvalg 49997	. . . . . 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 7429	. . . . . . . 8 ⊢ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ∈ V
11	10	mptex 7207	. . . . . . 7 ⊢ (𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑘 ∘_func 𝐹)) ∈ V
12	10, 10	mpoex 8060	. . . . . . 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 7981	. . . . . 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 5881	. . . 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 5751	. . 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 260	. 2 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → Rel dom (2^nd ‘(𝑃 −∘_F 𝐹)))
18		rel0 5771	. . 3 ⊢ Rel ∅
19		reldmprcof 49996	. . . . . . . . 9 ⊢ Rel dom −∘_F
20	19	ovprc 7434	. . . . . . . 8 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (𝑃 −∘_F 𝐹) = ∅)
21	20	fveq2d 6871	. . . . . . 7 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (2^nd ‘(𝑃 −∘_F 𝐹)) = (2^nd ‘∅))
22		2nd0 7977	. . . . . . 7 ⊢ (2^nd ‘∅) = ∅
23	21, 22	eqtrdi 2813	. . . . . 6 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (2^nd ‘(𝑃 −∘_F 𝐹)) = ∅)
24	23	dmeqd 5881	. . . . 5 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (2^nd ‘(𝑃 −∘_F 𝐹)) = dom ∅)
25		dm0 5896	. . . . 5 ⊢ dom ∅ = ∅
26	24, 25	eqtrdi 2813	. . . 4 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (2^nd ‘(𝑃 −∘_F 𝐹)) = ∅)
27	26	releqd 5751	. . 3 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (Rel dom (2^nd ‘(𝑃 −∘_F 𝐹)) ↔ Rel ∅))
28	18, 27	mpbiri 260	. 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 399 = wceq 1560 ∈ wcel 2142 Vcvv 3454 ∅c0 4285 ⟨cop 4588 ↦ cmpt 5181 dom cdm 5647 ∘ ccom 5651 Rel wrel 5652 ‘cfv 6521 (class class class)co 7396 ∈ cmpo 7398 1^st c1st 7968 2^nd c2nd 7969 Func cfunc 17887 ∘_func ccofu 17889 Nat cnat 17977 −∘_F cprcof 49994

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3077 df-rex 3087 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5542 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-ov 7399 df-oprab 7400 df-mpo 7401 df-1st 7970 df-2nd 7971 df-prcof 49995

This theorem is referenced by: (None)

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