prcofelvv - Mathbox for Zhi Wang

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Theorem prcofelvv 49878

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 2739	. . 3 ⊢ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) = ((1^st ‘𝑃) Func (2^nd ‘𝑃))
2		eqid 2739	. . 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 2740	. . 3 ⊢ (𝜑 → (1^st ‘𝑃) = (1^st ‘𝑃))
6		eqidd 2740	. . 3 ⊢ (𝜑 → (2^nd ‘𝑃) = (2^nd ‘𝑃))
7	1, 2, 3, 4, 5, 6	prcofvalg 49874	. 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 7390	. . . 4 ⊢ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ∈ V
9	8	mptex 7168	. . 3 ⊢ (𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑘 ∘_func 𝐹)) ∈ V
10	8, 8	mpoex 8022	. . 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 2847	1 ⊢ (𝜑 → (𝑃 −∘_F 𝐹) ∈ (V × V))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2119 Vcvv 3431 ⟨cop 4562 ↦ cmpt 5154 × cxp 5617 ∘ ccom 5623 ‘cfv 6486 (class class class)co 7357 ∈ cmpo 7359 1^st c1st 7930 2^nd c2nd 7931 Func cfunc 17813 ∘_func ccofu 17815 Nat cnat 17903 −∘_F cprcof 49871

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 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679

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 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4840 df-iun 4924 df-br 5074 df-opab 5136 df-mpt 5155 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 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7932 df-2nd 7933 df-prcof 49872

This theorem is referenced by: relran 50122 ranval3 50129 ranrcl4lem 50136 ranup 50140

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