prcofelvv - Mathbox for Zhi Wang

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Theorem prcofelvv 49359

Description: The pre-composition functor is an ordered pair. (Contributed by Zhi Wang, 4-Nov-2025.)

**Hypotheses**
Ref	Expression
prcofelvv.f	⊢ (𝜑 → 𝐹 ∈ 𝑈)
prcofelvv.p	⊢ (𝜑 → 𝑃 ∈ 𝑉)

**Assertion**
Ref	Expression
prcofelvv	⊢ (𝜑 → (𝑃 −∘_F 𝐹) ∈ (V × V))

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

Step	Hyp	Ref	Expression
1		eqid 2730	. . 3 ⊢ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) = ((1^st ‘𝑃) Func (2^nd ‘𝑃))
2		eqid 2730	. . 3 ⊢ ((1^st ‘𝑃) Nat (2^nd ‘𝑃)) = ((1^st ‘𝑃) Nat (2^nd ‘𝑃))
3		prcofelvv.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑈)
4		prcofelvv.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ 𝑉)
5		eqidd 2731	. . 3 ⊢ (𝜑 → (1^st ‘𝑃) = (1^st ‘𝑃))
6		eqidd 2731	. . 3 ⊢ (𝜑 → (2^nd ‘𝑃) = (2^nd ‘𝑃))
7	1, 2, 3, 4, 5, 6	prcofvalg 49355	. 2 ⊢ (𝜑 → (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))))⟩)
8		ovex 7422	. . . 4 ⊢ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ∈ V
9	8	mptex 7199	. . 3 ⊢ (𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑘 ∘_func 𝐹)) ∈ V
10	8, 8	mpoex 8060	. . 3 ⊢ (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
11	9, 10	opelvv 5680	. 2 ⊢ ⟨(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))))⟩ ∈ (V × V)
12	7, 11	eqeltrdi 2837	1 ⊢ (𝜑 → (𝑃 −∘_F 𝐹) ∈ (V × V))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2109 Vcvv 3450 ⟨cop 4597 ↦ cmpt 5190 × cxp 5638 ∘ ccom 5644 ‘cfv 6513 (class class class)co 7389 ∈ cmpo 7391 1^st c1st 7968 2^nd c2nd 7969 Func cfunc 17822 ∘_func ccofu 17824 Nat cnat 17912 −∘_F cprcof 49352

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-id 5535 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-ov 7392 df-oprab 7393 df-mpo 7394 df-1st 7970 df-2nd 7971 df-prcof 49353

This theorem is referenced by: relran 49603 ranval3 49610 ranrcl4lem 49617 ranup 49621

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