Theorem fucoelvv 49309

Description: A functor composition bifunctor is an ordered pair. Enables 1st2ndb 8008. (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) = ⚬ )

Assertion

Ref	Expression
fucoelvv	⊢ (𝜑 → ⚬ ∈ (V × V))

Proof of Theorem fucoelvv

Dummy variables 𝑎 𝑏 𝑓 𝑔 𝑘 𝑙 𝑚 𝑟 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucofval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑇)
2		fucofval.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑈)
3		fucofval.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
4		fucofval.o	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⚬ )
5		eqidd 2730	. . 3 ⊢ (𝜑 → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
6	1, 2, 3, 4, 5	fucofval 49308	. 2 ⊢ (𝜑 → ⚬ = ⟨( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))), (𝑢 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)), 𝑣 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝐷 Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝐶 Nat 𝐷)(2nd ‘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))))))⟩)
7		df-cofu 17822	. . . . 5 ⊢ ∘func = (𝑔 ∈ V, 𝑓 ∈ V ↦ ⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩)
8	7	mpofun 7513	. . . 4 ⊢ Fun ∘func
9		ovex 7420	. . . . 5 ⊢ (𝐷 Func 𝐸) ∈ V
10		ovex 7420	. . . . 5 ⊢ (𝐶 Func 𝐷) ∈ V
11	9, 10	xpex 7729	. . . 4 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∈ V
12		resfunexg 7189	. . . 4 ⊢ ((Fun ∘func ∧ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∈ V) → ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ∈ V)
13	8, 11, 12	mp2an 692	. . 3 ⊢ ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ∈ V
14	11, 11	mpoex 8058	. . 3 ⊢ (𝑢 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)), 𝑣 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝐷 Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝐶 Nat 𝐷)(2nd ‘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
15	13, 14	opelvv 5678	. 2 ⊢ ⟨( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))), (𝑢 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)), 𝑣 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝐷 Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝐶 Nat 𝐷)(2nd ‘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 × V)
16	6, 15	eqeltrdi 2836	1 ⊢ (𝜑 → ⚬ ∈ (V × V))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ⦋csb 3862  ⟨cop 4595   ↦ cmpt 5188   × cxp 5636  dom cdm 5638   ↾ cres 5640   ∘ ccom 5642  Fun wfun 6505  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  1^st c1st 7966  2^nd c2nd 7967  Basecbs 17179  compcco 17232   Func cfunc 17816   ∘_func ccofu 17818   Nat cnat 17906   ∘_F cfuco 49305

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-cofu 17822  df-fuco 49306

This theorem is referenced by:  fuco1  49310  fuco2  49312  fuco11b  49326  fuco11bALT  49327  fucofunca  49349  fucolid  49350  fucorid  49351  precofvalALT  49357