prcofelvv - Mathbox for Zhi Wang

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Theorem prcofelvv 49375

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 2729	. . 3 ⊢ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) = ((1^st ‘𝑃) Func (2^nd ‘𝑃))
2		eqid 2729	. . 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 2730	. . 3 ⊢ (𝜑 → (1^st ‘𝑃) = (1^st ‘𝑃))
6		eqidd 2730	. . 3 ⊢ (𝜑 → (2^nd ‘𝑃) = (2^nd ‘𝑃))
7	1, 2, 3, 4, 5, 6	prcofvalg 49371	. 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 7382	. . . 4 ⊢ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ∈ V
9	8	mptex 7159	. . 3 ⊢ (𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑘 ∘_func 𝐹)) ∈ V
10	8, 8	mpoex 8014	. . 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 5659	. 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 2836	1 ⊢ (𝜑 → (𝑃 −∘_F 𝐹) ∈ (V × V))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2109 Vcvv 3436 ⟨cop 4583 ↦ cmpt 5173 × cxp 5617 ∘ ccom 5623 ‘cfv 6482 (class class class)co 7349 ∈ cmpo 7351 1^st c1st 7922 2^nd c2nd 7923 Func cfunc 17761 ∘_func ccofu 17763 Nat cnat 17851 −∘_F cprcof 49368

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 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-ov 7352 df-oprab 7353 df-mpo 7354 df-1st 7924 df-2nd 7925 df-prcof 49369

This theorem is referenced by: relran 49619 ranval3 49626 ranrcl4lem 49633 ranup 49637

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