Theorem fucofulem2 49896

Description: Lemma for proving functor theorems. Maybe consider eufnfv 7209 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 2761	. . . . 5 ⊢ ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷)) = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))
3	2	xpcfucbas 49837	. . . 4 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = (Base‘((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷)))
4	1, 3	eqtri 2784	. . 3 ⊢ 𝐵 = (Base‘((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷)))
5	4	funcf2lem2 49667	. 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 7523	. . 3 ⊢ (𝐺 Fn (𝐵 × 𝐵) ↔ 𝐺 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑢𝐺𝑣)))
7		ffnfv 7096	. . . . . . 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 486	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → 𝑚 ∈ 𝐵)
10		simpr 488	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → 𝑛 ∈ 𝐵)
11	2, 4, 8, 9, 10	xpcfuchom 49839	. . . . . . . . . 10 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → (𝑚𝐻𝑛) = (((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)) × ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))))
12	11	fneq2d 6611	. . . . . . . . 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 7523	. . . . . . . . 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 289	. . . . . . . 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 3319	. . . . . . . . 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 6863	. . . . . . . . . . . 12 ⊢ (𝑟 = ⟨𝑝, 𝑞⟩ → ((𝑚𝐺𝑛)‘𝑟) = ((𝑚𝐺𝑛)‘⟨𝑝, 𝑞⟩))
17		df-ov 7395	. . . . . . . . . . . 12 ⊢ (𝑝(𝑚𝐺𝑛)𝑞) = ((𝑚𝐺𝑛)‘⟨𝑝, 𝑞⟩)
18	16, 17	eqtr4di 2814	. . . . . . . . . . 11 ⊢ (𝑟 = ⟨𝑝, 𝑞⟩ → ((𝑚𝐺𝑛)‘𝑟) = (𝑝(𝑚𝐺𝑛)𝑞))
19	18	eleq1d 2846	. . . . . . . . . 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 5811	. . . . . . . . 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 289	. . . . . . . 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 641	. . . . . . 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 285	. . . . . 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 485	. . . . 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 3223	. . . 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 1566	. . 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 637	. 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 277	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)))))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1559  ⊤wtru 1560   ∈ wcel 2141  ∀wral 3075  ⟨cop 4587   × cxp 5643   Fn wfn 6512  ⟶wf 6513  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394  1^st c1st 7964  2^nd c2nd 7965   ↑_m cmap 8803  Xcixp 8875  Basecbs 17228  Hom chom 17280   Func cfunc 17870   Nat cnat 17960   FuncCat cfuc 17961   ×_c cxpc 18183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8673  df-map 8805  df-ixp 8876  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12479  df-z 12566  df-dec 12686  df-uz 12837  df-fz 13510  df-struct 17166  df-slot 17201  df-ndx 17213  df-base 17229  df-hom 17293  df-cco 17294  df-func 17874  df-nat 17962  df-fuc 17963  df-xpc 18187

This theorem is referenced by: (None)