fucofn2 - Mathbox for Zhi Wang

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Theorem fucofn2 49219

Description: The morphism part of the functor composition bifunctor is a function on the Cartesian square of the base set. (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)
fuco1.o	⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
fuco1.w	⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))

**Assertion**
Ref	Expression
fucofn2	⊢ (𝜑 → 𝑃 Fn (𝑊 × 𝑊))

Proof of Theorem fucofn2 Dummy variables 𝑎 𝑏 𝑓 𝑘 𝑙 𝑚 𝑟 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2730	. . 3 ⊢ (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)))))) = (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))))))
2		ovex 7427	. . . . . . . . 9 ⊢ ((1^st ‘𝑢)(𝐷 Nat 𝐸)(1^st ‘𝑣)) ∈ V
3		ovex 7427	. . . . . . . . 9 ⊢ ((2^nd ‘𝑢)(𝐶 Nat 𝐷)(2^nd ‘𝑣)) ∈ V
4	2, 3	mpoex 8067	. . . . . . . 8 ⊢ (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))))) ∈ V
5	4	csbex 5274	. . . . . . 7 ⊢ ⦋(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))))) ∈ V
6	5	csbex 5274	. . . . . 6 ⊢ ⦋(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))))) ∈ V
7	6	csbex 5274	. . . . 5 ⊢ ⦋(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))))) ∈ V
8	7	csbex 5274	. . . 4 ⊢ ⦋(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))))) ∈ V
9	8	csbex 5274	. . 3 ⊢ ⦋(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))))) ∈ V
10	1, 9	fnmpoi 8058	. 2 ⊢ (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)))))) Fn (𝑊 × 𝑊)
11		fucofval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑇)
12		fucofval.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑈)
13		fucofval.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
14		fuco1.o	. . . 4 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
15		fuco1.w	. . . 4 ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
16	11, 12, 13, 14, 15	fuco2 49218	. . 3 ⊢ (𝜑 → 𝑃 = (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)))))))
17	16	fneq1d 6619	. 2 ⊢ (𝜑 → (𝑃 Fn (𝑊 × 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)))))) Fn (𝑊 × 𝑊)))
18	10, 17	mpbiri 258	1 ⊢ (𝜑 → 𝑃 Fn (𝑊 × 𝑊))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ⦋csb 3870 ⟨cop 4603 ↦ cmpt 5196 × cxp 5644 Fn wfn 6514 ‘cfv 6519 (class class class)co 7394 ∈ cmpo 7396 1^st c1st 7975 2^nd c2nd 7976 Basecbs 17185 compcco 17238 Func cfunc 17822 Nat cnat 17912 ∘_F cfuco 49211

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5242 ax-sep 5259 ax-nul 5269 ax-pow 5328 ax-pr 5395 ax-un 7718

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2880 df-ne 2928 df-ral 3047 df-rex 3056 df-reu 3358 df-rab 3412 df-v 3457 df-sbc 3762 df-csb 3871 df-dif 3925 df-un 3927 df-in 3929 df-ss 3939 df-nul 4305 df-if 4497 df-pw 4573 df-sn 4598 df-pr 4600 df-op 4604 df-uni 4880 df-iun 4965 df-br 5116 df-opab 5178 df-mpt 5197 df-id 5541 df-xp 5652 df-rel 5653 df-cnv 5654 df-co 5655 df-dm 5656 df-rn 5657 df-res 5658 df-ima 5659 df-iota 6472 df-fun 6521 df-fn 6522 df-f 6523 df-f1 6524 df-fo 6525 df-f1o 6526 df-fv 6527 df-ov 7397 df-oprab 7398 df-mpo 7399 df-1st 7977 df-2nd 7978 df-cofu 17828 df-fuco 49212

This theorem is referenced by: fucofunc 49254

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