Theorem fucofulem2 50245

Description: Lemma for proving functor theorems. Maybe consider eufnfv 7232 to prove the uniqueness of a functor. (Contributed by Zhi Wang, 25-Sep-2025.)

Hypotheses

Ref	Expression
fucofulem2.b	⊢ 𝐵 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))
fucofulem2.h	⊢ 𝐻 = (Hom ‘((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷)))

Assertion

Ref	Expression
fucofulem2	⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))(𝐶 Nat 𝐸)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑢𝐺𝑣)) ∧ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 ((𝑚𝐺𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)))))

Distinct variable groups:   𝐵,𝑚,𝑛,𝑧   𝑢,𝐵,𝑣   𝐶,𝑎,𝑏,𝑚,𝑛   𝐶,𝑝,𝑞,𝑚,𝑛   𝑧,𝐶   𝐷,𝑎,𝑏   𝐷,𝑝,𝑞   𝐸,𝑎,𝑏,𝑚,𝑛   𝐸,𝑝,𝑞   𝑧,𝐸   𝑚,𝐹,𝑛,𝑝,𝑞   𝑧,𝐹   𝐺,𝑎,𝑏,𝑚,𝑛   𝐺,𝑝,𝑞   𝑢,𝐺,𝑣   𝑧,𝐺   𝑚,𝐻,𝑛,𝑧

Allowed substitution hints:   𝐵(𝑞, 𝑝, 𝑎, 𝑏)   𝐶(𝑣, 𝑢)   𝐷(𝑧, 𝑣, 𝑢, 𝑚, 𝑛)   𝐸(𝑣, 𝑢)   𝐹(𝑣, 𝑢, 𝑎, 𝑏)   𝐻(𝑣, 𝑢, 𝑞, 𝑝, 𝑎, 𝑏)

Proof of Theorem fucofulem2

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucofulem2.b	. . . 4 ⊢ 𝐵 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))
2		eqid 2762	. . . . 5 ⊢ ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷)) = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))
3	2	xpcfucbas 50186	. . . 4 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = (Base‘((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷)))
4	1, 3	eqtri 2785	. . 3 ⊢ 𝐵 = (Base‘((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷)))
5	4	funcf2lem2 50016	. 2 ⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))(𝐶 Nat 𝐸)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 (𝑚𝐺𝑛):(𝑚𝐻𝑛)⟶((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))))
6		fnov 7548	. . 3 ⊢ (𝐺 Fn (𝐵 × 𝐵) ↔ 𝐺 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑢𝐺𝑣)))
7		ffnfv 7116	. . . . . . 7 ⊢ ((𝑚𝐺𝑛):(𝑚𝐻𝑛)⟶((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ ((𝑚𝐺𝑛) Fn (𝑚𝐻𝑛) ∧ ∀𝑟 ∈ (𝑚𝐻𝑛)((𝑚𝐺𝑛)‘𝑟) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))))
8		fucofulem2.h	. . . . . . . . . . 11 ⊢ 𝐻 = (Hom ‘((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷)))
9		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → 𝑚 ∈ 𝐵)
10		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → 𝑛 ∈ 𝐵)
11	2, 4, 8, 9, 10	xpcfuchom 50188	. . . . . . . . . 10 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → (𝑚𝐻𝑛) = (((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)) × ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))))
12	11	fneq2d 6630	. . . . . . . . 9 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → ((𝑚𝐺𝑛) Fn (𝑚𝐻𝑛) ↔ (𝑚𝐺𝑛) Fn (((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)) × ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)))))
13		fnov 7548	. . . . . . . . 9 ⊢ ((𝑚𝐺𝑛) Fn (((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)) × ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))) ↔ (𝑚𝐺𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)))
14	12, 13	bitrdi 290	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → ((𝑚𝐺𝑛) Fn (𝑚𝐻𝑛) ↔ (𝑚𝐺𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎))))
15	11	raleqdv 3321	. . . . . . . . 9 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → (∀𝑟 ∈ (𝑚𝐻𝑛)((𝑚𝐺𝑛)‘𝑟) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ ∀𝑟 ∈ (((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)) × ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)))((𝑚𝐺𝑛)‘𝑟) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))))
16		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑟 = ⟨𝑝, 𝑞⟩ → ((𝑚𝐺𝑛)‘𝑟) = ((𝑚𝐺𝑛)‘⟨𝑝, 𝑞⟩))
17		df-ov 7420	. . . . . . . . . . . 12 ⊢ (𝑝(𝑚𝐺𝑛)𝑞) = ((𝑚𝐺𝑛)‘⟨𝑝, 𝑞⟩)
18	16, 17	eqtr4di 2815	. . . . . . . . . . 11 ⊢ (𝑟 = ⟨𝑝, 𝑞⟩ → ((𝑚𝐺𝑛)‘𝑟) = (𝑝(𝑚𝐺𝑛)𝑞))
19	18	eleq1d 2847	. . . . . . . . . 10 ⊢ (𝑟 = ⟨𝑝, 𝑞⟩ → (((𝑚𝐺𝑛)‘𝑟) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ (𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))))
20	19	ralxp 5825	. . . . . . . . 9 ⊢ (∀𝑟 ∈ (((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)) × ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)))((𝑚𝐺𝑛)‘𝑟) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)))
21	15, 20	bitrdi 290	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → (∀𝑟 ∈ (𝑚𝐻𝑛)((𝑚𝐺𝑛)‘𝑟) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))))
22	14, 21	anbi12d 644	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → (((𝑚𝐺𝑛) Fn (𝑚𝐻𝑛) ∧ ∀𝑟 ∈ (𝑚𝐻𝑛)((𝑚𝐺𝑛)‘𝑟) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))) ↔ ((𝑚𝐺𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)))))
23	7, 22	bitrid 286	. . . . . 6 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → ((𝑚𝐺𝑛):(𝑚𝐻𝑛)⟶((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ ((𝑚𝐺𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)))))
24	23	adantl 487	. . . . 5 ⊢ ((⊤ ∧ (𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵)) → ((𝑚𝐺𝑛):(𝑚𝐻𝑛)⟶((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ ((𝑚𝐺𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)))))
25	24	2ralbidva 3226	. . . 4 ⊢ (⊤ → (∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 (𝑚𝐺𝑛):(𝑚𝐻𝑛)⟶((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 ((𝑚𝐺𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)))))
26	25	mptru 1577	. . 3 ⊢ (∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 (𝑚𝐺𝑛):(𝑚𝐻𝑛)⟶((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 ((𝑚𝐺𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))))
27	6, 26	anbi12i 640	. 2 ⊢ ((𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 (𝑚𝐺𝑛):(𝑚𝐻𝑛)⟶((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))) ↔ (𝐺 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑢𝐺𝑣)) ∧ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 ((𝑚𝐺𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)))))
28	5, 27	bitri 278	1 ⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))(𝐶 Nat 𝐸)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑢𝐺𝑣)) ∧ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 ((𝑚𝐺𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)))))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ⊤wtru 1571   ∈ wcel 2145  ∀wral 3078  ⟨cop 4593   × cxp 5657   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  1^st c1st 7988  2^nd c2nd 7989   ↑_m cmap 8830  Xcixp 8908  Basecbs 17307  Hom chom 17359   Func cfunc 17949   Nat cnat 18039   FuncCat cfuc 18040   ×_c cxpc 18262

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-ixp 8909  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-2 12331  df-3 12332  df-4 12333  df-5 12334  df-6 12335  df-7 12336  df-8 12337  df-9 12338  df-n0 12533  df-z 12620  df-dec 12741  df-uz 12892  df-fz 13566  df-struct 17245  df-slot 17280  df-ndx 17292  df-base 17308  df-hom 17372  df-cco 17373  df-func 17953  df-nat 18041  df-fuc 18042  df-xpc 18266

This theorem is used by: (None)