fucofval - Mathbox for Zhi Wang

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Theorem fucofval 50074

Description: Value of the function giving the functor composition bifunctor. Hypotheses fucofval.c and fucofval.d are not redundant (fucofvalne 50080). (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 5447	. . 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 7999	. . 3 ⊢ ((𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈) → (1^st ‘⟨𝐶, 𝐷⟩) = 𝐶)
6	3, 4, 5	syl2anc 595	. 2 ⊢ (𝜑 → (1^st ‘⟨𝐶, 𝐷⟩) = 𝐶)
7		op2ndg 8000	. . 3 ⊢ ((𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈) → (2^nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
8	3, 4, 7	syl2anc 595	. 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 50073	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 1570 ∈ wcel 2143 Vcvv 3455 ⦋csb 3854 ⟨cop 4596 ↦ cmpt 5193 × cxp 5661 ↾ cres 5665 ‘cfv 6538 (class class class)co 7412 ∈ cmpo 7414 1^st c1st 7985 2^nd c2nd 7986 Basecbs 17270 compcco 17323 Func cfunc 17912 ∘_func ccofu 17914 Nat cnat 18002 ∘_F cfuco 50071

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-iota 6494 df-fun 6540 df-fv 6546 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7987 df-2nd 7988 df-fuco 50072

This theorem is referenced by: fucoelvv 50075 fuco1 50076 fuco2 50078

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