prcofelvv - Mathbox for Zhi Wang

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Theorem prcofelvv 50487

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 2761	. . 3 ⊢ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) = ((1^st ‘𝑃) Func (2^nd ‘𝑃))
2		eqid 2761	. . 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 2762	. . 3 ⊢ (𝜑 → (1^st ‘𝑃) = (1^st ‘𝑃))
6		eqidd 2762	. . 3 ⊢ (𝜑 → (2^nd ‘𝑃) = (2^nd ‘𝑃))
7	1, 2, 3, 4, 5, 6	prcofvalg 50483	. 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 7453	. . . 4 ⊢ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ∈ V
9	8	mptex 7229	. . 3 ⊢ (𝑘 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)) ↦ (𝑘 ∘_func 𝐹)) ∈ V
10	8, 8	mpoex 8092	. . 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 5691	. 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 2869	1 ⊢ (𝜑 → (𝑃 −∘_F 𝐹) ∈ (V × V))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∈ wcel 2145 Vcvv 3451 ⟨cop 4590 ↦ cmpt 5186 × cxp 5649 ∘ ccom 5655 ‘cfv 6538 (class class class)co 7420 ∈ cmpo 7422 1^st c1st 7999 2^nd c2nd 8000 Func cfunc 18029 ∘_func ccofu 18031 Nat cnat 18119 −∘_F cprcof 50480

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 8001 df-2nd 8002 df-prcof 50481

This theorem is used by: relran 50731 ranval3 50738 ranrcl4lem 50745 ranup 50749

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