Theorem fucoelvv 49817

Description: A functor composition bifunctor is an ordered pair. Enables 1st2ndb 7978. (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 2741	. . 3 ⊢ (𝜑 → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
6	1, 2, 3, 4, 5	fucofval 49816	. 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 17825	. . . . 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 7487	. . . 4 ⊢ Fun ∘func
9		ovex 7396	. . . . 5 ⊢ (𝐷 Func 𝐸) ∈ V
10		ovex 7396	. . . . 5 ⊢ (𝐶 Func 𝐷) ∈ V
11	9, 10	xpex 7703	. . . 4 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∈ V
12		resfunexg 7166	. . . 4 ⊢ ((Fun ∘func ∧ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∈ V) → ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ∈ V)
13	8, 11, 12	mp2an 698	. . 3 ⊢ ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ∈ V
14	11, 11	mpoex 8028	. . 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 5665	. 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 2848	1 ⊢ (𝜑 → ⚬ ∈ (V × V))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  Vcvv 3432  ⦋csb 3838  ⟨cop 4568   ↦ cmpt 5160   × cxp 5623  dom cdm 5625   ↾ cres 5627   ∘ ccom 5629  Fun wfun 6486  ‘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-rep 5206  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-reu 3346  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-iun 4930  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-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-cofu 17825  df-fuco 49814

This theorem is referenced by:  fuco1  49818  fuco2  49820  fuco11b  49834  fuco11bALT  49835  fucofunca  49857  fucolid  49858  fucorid  49859  precofvalALT  49865