fucofval - Mathbox for Zhi Wang

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Theorem fucofval 49806

Description: Value of the function giving the functor composition bifunctor. Hypotheses fucofval.c and fucofval.d are not redundant (fucofvalne 49812). (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 5411	. . 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 7947	. . 3 ⊢ ((𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈) → (1^st ‘⟨𝐶, 𝐷⟩) = 𝐶)
6	3, 4, 5	syl2anc 585	. 2 ⊢ (𝜑 → (1^st ‘⟨𝐶, 𝐷⟩) = 𝐶)
7		op2ndg 7948	. . 3 ⊢ ((𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈) → (2^nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
8	3, 4, 7	syl2anc 585	. 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 49805	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 1542 ∈ wcel 2114 Vcvv 3430 ⦋csb 3838 ⟨cop 4574 ↦ cmpt 5167 × cxp 5622 ↾ cres 5626 ‘cfv 6492 (class class class)co 7360 ∈ cmpo 7362 1^st c1st 7933 2^nd c2nd 7934 Basecbs 17170 compcco 17223 Func cfunc 17812 ∘_func ccofu 17814 Nat cnat 17902 ∘_F cfuco 49803

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-iota 6448 df-fun 6494 df-fv 6500 df-ov 7363 df-oprab 7364 df-mpo 7365 df-1st 7935 df-2nd 7936 df-fuco 49804

This theorem is referenced by: fucoelvv 49807 fuco1 49808 fuco2 49810

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