fucofval - Mathbox for Zhi Wang

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Theorem fucofval 49816

Description: Value of the function giving the functor composition bifunctor. Hypotheses fucofval.c and fucofval.d are not redundant (fucofvalne 49822). (Contributed by Zhi Wang, 29-Sep-2025.)

**Hypotheses**
Ref	Expression
fucofval.c	⊢ (𝜑 → 𝐶 ∈ 𝑇)
fucofval.d	⊢ (𝜑 → 𝐷 ∈ 𝑈)
fucofval.e	⊢ (𝜑 → 𝐸 ∈ 𝑉)
fucofval.o	⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⚬ )
fucofval.w	⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))

**Assertion**
Ref	Expression
fucofval	⊢ (𝜑 → ⚬ = ⟨( ∘_func ↾ 𝑊), (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ ⦋(1^st ‘(2^nd ‘𝑢)) / 𝑓⦌⦋(1^st ‘(1^st ‘𝑢)) / 𝑘⦌⦋(2^nd ‘(1^st ‘𝑢)) / 𝑙⦌⦋(1^st ‘(2^nd ‘𝑣)) / 𝑚⦌⦋(1^st ‘(1^st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1^st ‘𝑢)(𝐷 Nat 𝐸)(1^st ‘𝑣)), 𝑎 ∈ ((2^nd ‘𝑢)(𝐶 Nat 𝐷)(2^nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝐸)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩)

Distinct variable groups: 𝐶,𝑎,𝑏,𝑓,𝑘,𝑙,𝑚,𝑟,𝑢,𝑣,𝑥 𝐷,𝑎,𝑏,𝑓,𝑘,𝑙,𝑚,𝑟,𝑢,𝑣,𝑥 𝐸,𝑎,𝑏,𝑓,𝑘,𝑙,𝑚,𝑟,𝑢,𝑣,𝑥 𝑊,𝑎,𝑏,𝑓,𝑘,𝑙,𝑚,𝑟,𝑢,𝑣,𝑥 𝜑,𝑎,𝑏,𝑓,𝑘,𝑙,𝑚,𝑟,𝑢,𝑣,𝑥 Allowed substitution hints: 𝑇(𝑥,𝑣,𝑢,𝑓,𝑘,𝑚,𝑟,𝑎,𝑏,𝑙) 𝑈(𝑥,𝑣,𝑢,𝑓,𝑘,𝑚,𝑟,𝑎,𝑏,𝑙) 𝑉(𝑥,𝑣,𝑢,𝑓,𝑘,𝑚,𝑟,𝑎,𝑏,𝑙) ⚬ (𝑥,𝑣,𝑢,𝑓,𝑘,𝑚,𝑟,𝑎,𝑏,𝑙)

**Proof of Theorem fucofval**
Step	Hyp	Ref	Expression
1		opex 5410	. . 3 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
2	1	a1i 11	. 2 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ V)
3		fucofval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑇)
4		fucofval.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑈)
5		op1stg 7950	. . 3 ⊢ ((𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈) → (1^st ‘⟨𝐶, 𝐷⟩) = 𝐶)
6	3, 4, 5	syl2anc 590	. 2 ⊢ (𝜑 → (1^st ‘⟨𝐶, 𝐷⟩) = 𝐶)
7		op2ndg 7951	. . 3 ⊢ ((𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈) → (2^nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
8	3, 4, 7	syl2anc 590	. 2 ⊢ (𝜑 → (2^nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
9		fucofval.e	. 2 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
10		fucofval.o	. 2 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⚬ )
11		fucofval.w	. 2 ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
12	2, 6, 8, 9, 10, 11	fucofvalg 49815	1 ⊢ (𝜑 → ⚬ = ⟨( ∘_func ↾ 𝑊), (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ ⦋(1^st ‘(2^nd ‘𝑢)) / 𝑓⦌⦋(1^st ‘(1^st ‘𝑢)) / 𝑘⦌⦋(2^nd ‘(1^st ‘𝑢)) / 𝑙⦌⦋(1^st ‘(2^nd ‘𝑣)) / 𝑚⦌⦋(1^st ‘(1^st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1^st ‘𝑢)(𝐷 Nat 𝐸)(1^st ‘𝑣)), 𝑎 ∈ ((2^nd ‘𝑢)(𝐶 Nat 𝐷)(2^nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝐸)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1547 ∈ wcel 2119 Vcvv 3432 ⦋csb 3838 ⟨cop 4568 ↦ cmpt 5160 × cxp 5623 ↾ cres 5627 ‘cfv 6492 (class class class)co 7363 ∈ cmpo 7365 1^st c1st 7936 2^nd c2nd 7937 Basecbs 17177 compcco 17230 Func cfunc 17819 ∘_func ccofu 17821 Nat cnat 17909 ∘_F cfuco 49813

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 2712 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685

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 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-iota 6448 df-fun 6494 df-fv 6500 df-ov 7366 df-oprab 7367 df-mpo 7368 df-1st 7938 df-2nd 7939 df-fuco 49814

This theorem is referenced by: fucoelvv 49817 fuco1 49818 fuco2 49820

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