Theorem fucoelvv 49351

Description: A functor composition bifunctor is an ordered pair. Enables 1st2ndb 7961. (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 2732	. . 3 ⊢ (𝜑 → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
6	1, 2, 3, 4, 5	fucofval 49350	. 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 17764	. . . . 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 7470	. . . 4 ⊢ Fun ∘func
9		ovex 7379	. . . . 5 ⊢ (𝐷 Func 𝐸) ∈ V
10		ovex 7379	. . . . 5 ⊢ (𝐶 Func 𝐷) ∈ V
11	9, 10	xpex 7686	. . . 4 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∈ V
12		resfunexg 7149	. . . 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 8011	. . 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 5656	. 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 2839	1 ⊢ (𝜑 → ⚬ ∈ (V × V))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⦋csb 3850  ⟨cop 4582   ↦ cmpt 5172   × cxp 5614  dom cdm 5616   ↾ cres 5618   ∘ ccom 5620  Fun wfun 6475  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  1^st c1st 7919  2^nd c2nd 7920  Basecbs 17117  compcco 17170   Func cfunc 17758   ∘_func ccofu 17760   Nat cnat 17848   ∘_F cfuco 49347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-cofu 17764  df-fuco 49348

This theorem is referenced by:  fuco1  49352  fuco2  49354  fuco11b  49368  fuco11bALT  49369  fucofunca  49391  fucolid  49392  fucorid  49393  precofvalALT  49399