Theorem fucoelvv 48967

Description: A functor composition bifunctor is an ordered pair. Enables 1st2ndb 8035. (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 2735	. . 3 ⊢ (𝜑 → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
6	1, 2, 3, 4, 5	fucofval 48966	. 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 17875	. . . . 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 7538	. . . 4 ⊢ Fun ∘func
9		ovex 7445	. . . . 5 ⊢ (𝐷 Func 𝐸) ∈ V
10		ovex 7445	. . . . 5 ⊢ (𝐶 Func 𝐷) ∈ V
11	9, 10	xpex 7754	. . . 4 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∈ V
12		resfunexg 7216	. . . 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 8085	. . 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 5705	. 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 2841	1 ⊢ (𝜑 → ⚬ ∈ (V × V))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  Vcvv 3463  ⦋csb 3879  ⟨cop 4612   ↦ cmpt 5205   × cxp 5663  dom cdm 5665   ↾ cres 5667   ∘ ccom 5669  Fun wfun 6534  ‘cfv 6540  (class class class)co 7412   ∈ cmpo 7414  1^st c1st 7993  2^nd c2nd 7994  Basecbs 17228  compcco 17284   Func cfunc 17869   ∘_func ccofu 17871   Nat cnat 17959   ∘_F cfuco 48963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7736

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6493  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7995  df-2nd 7996  df-cofu 17875  df-fuco 48964

This theorem is referenced by:  fuco1  48968  fuco2  48970  fuco11b  48984  fuco11bALT  48985  fucofunca  49007  fucolid  49008  fucorid  49009  precofvalALT  49015