Theorem fucoelvv 50535

Description: A functor composition bifunctor is an ordered pair. Enables 1st2ndb 8041. (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 2761	. . 3 ⊢ (𝜑 → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
6	1, 2, 3, 4, 5	fucofval 50534	. 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 18040	. . . . 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 7544	. . . 4 ⊢ Fun ∘func
9		ovex 7453	. . . . 5 ⊢ (𝐷 Func 𝐸) ∈ V
10		ovex 7453	. . . . 5 ⊢ (𝐶 Func 𝐷) ∈ V
11	9, 10	xpex 7767	. . . 4 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∈ V
12		resfunexg 7221	. . . 4 ⊢ ((Fun ∘func ∧ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∈ V) → ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ∈ V)
13	8, 11, 12	mp2an 705	. . 3 ⊢ ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ∈ V
14	11, 11	mpoex 8092	. . 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 5690	. 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 2868	1 ⊢ (𝜑 → ⚬ ∈ (V × V))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ⦋csb 3846  ⟨cop 4834   ↦ cmpt 5185   × cxp 5648  dom cdm 5650   ↾ cres 5652   ∘ ccom 5654  Fun wfun 6532  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17392  compcco 17445   Func cfunc 18034   ∘_func ccofu 18036   Nat cnat 18124   ∘_F cfuco 50531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-cofu 18040  df-fuco 50532

This theorem is used by:  fuco1  50536  fuco2  50538  fuco11b  50552  fuco11bALT  50553  fucofunca  50575  fucolid  50576  fucorid  50577  precofvalALT  50596