prcofelvv - Mathbox for Zhi Wang

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Theorem prcofelvv 50660

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 2763	. . 3 ⊢ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) = ((1^st ‘𝑃) Func (2^nd ‘𝑃))
2		eqid 2763	. . 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 2764	. . 3 ⊢ (𝜑 → (1^st ‘𝑃) = (1^st ‘𝑃))
6		eqidd 2764	. . 3 ⊢ (𝜑 → (2^nd ‘𝑃) = (2^nd ‘𝑃))
7	1, 2, 3, 4, 5, 6	prcofvalg 50656	. 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 7463	. . . 4 ⊢ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ∈ V
9	8	mptex 7239	. . 3 ⊢ (𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑘 ∘_func 𝐹)) ∈ V
10	8, 8	mpoex 8103	. . 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 5695	. 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 2872	1 ⊢ (𝜑 → (𝑃 −∘_F 𝐹) ∈ (V × V))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∈ wcel 2146 Vcvv 3454 ⟨cop 4838 ↦ cmpt 5189 × cxp 5653 ∘ ccom 5659 ‘cfv 6547 (class class class)co 7430 ∈ cmpo 7432 1^st c1st 8009 2^nd c2nd 8010 Func cfunc 18044 ∘_func ccofu 18046 Nat cnat 18134 −∘_F cprcof 50653

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8011 df-2nd 8012 df-prcof 50654

This theorem is used by: relran 50952 ranval3 50959 ranrcl4lem 50966 ranup 50970

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