fucoppcco - Mathbox for Zhi Wang

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Theorem fucoppcco 49441

Description: The opposite category of functors is compatible with the category of opposite functors in terms of composition. (Contributed by Zhi Wang, 18-Nov-2025.)

**Hypotheses**
Ref	Expression
fucoppc.o	⊢ 𝑂 = (oppCat‘𝐶)
fucoppc.p	⊢ 𝑃 = (oppCat‘𝐷)
fucoppc.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
fucoppc.r	⊢ 𝑅 = (oppCat‘𝑄)
fucoppc.s	⊢ 𝑆 = (𝑂 FuncCat 𝑃)
fucoppc.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
fucoppc.f	⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
fucoppc.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))
fucoppcco.a	⊢ (𝜑 → 𝐴 ∈ (𝑋(Hom ‘𝑅)𝑌))
fucoppcco.b	⊢ (𝜑 → 𝐵 ∈ (𝑌(Hom ‘𝑅)𝑍))

**Assertion**
Ref	Expression
fucoppcco	⊢ (𝜑 → ((𝑋𝐺𝑍)‘(𝐵(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐴)) = (((𝑌𝐺𝑍)‘𝐵)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐴)))

Distinct variable groups: 𝑥,𝑁,𝑦 𝑥,𝑋,𝑦 𝑥,𝑌,𝑦 𝑥,𝑍,𝑦 𝜑,𝑥,𝑦 Allowed substitution hints: 𝐴(𝑥,𝑦) 𝐵(𝑥,𝑦) 𝐶(𝑥,𝑦) 𝐷(𝑥,𝑦) 𝑃(𝑥,𝑦) 𝑄(𝑥,𝑦) 𝑅(𝑥,𝑦) 𝑆(𝑥,𝑦) 𝐹(𝑥,𝑦) 𝐺(𝑥,𝑦) 𝑂(𝑥,𝑦)

Proof of Theorem fucoppcco Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucoppc.s	. . 3 ⊢ 𝑆 = (𝑂 FuncCat 𝑃)
2		eqid 2731	. . 3 ⊢ (𝑂 Nat 𝑃) = (𝑂 Nat 𝑃)
3		fucoppc.o	. . . 4 ⊢ 𝑂 = (oppCat‘𝐶)
4		eqid 2731	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	3, 4	oppcbas 17619	. . 3 ⊢ (Base‘𝐶) = (Base‘𝑂)
6		eqid 2731	. . 3 ⊢ (comp‘𝑃) = (comp‘𝑃)
7		eqid 2731	. . 3 ⊢ (comp‘𝑆) = (comp‘𝑆)
8		fucoppcco.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝑋(Hom ‘𝑅)𝑌))
9		fucoppc.q	. . . . . . 7 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
10		fucoppc.n	. . . . . . 7 ⊢ 𝑁 = (𝐶 Nat 𝐷)
11	9, 10	fuchom 17866	. . . . . 6 ⊢ 𝑁 = (Hom ‘𝑄)
12		fucoppc.r	. . . . . 6 ⊢ 𝑅 = (oppCat‘𝑄)
13	11, 12	oppchom 17616	. . . . 5 ⊢ (𝑋(Hom ‘𝑅)𝑌) = (𝑌𝑁𝑋)
14	8, 13	eleqtrdi 2841	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝑌𝑁𝑋))
15		fucoppc.p	. . . . 5 ⊢ 𝑃 = (oppCat‘𝐷)
16		fucoppc.f	. . . . 5 ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
17	10	natrcl 17855	. . . . . . 7 ⊢ (𝐴 ∈ (𝑌𝑁𝑋) → (𝑌 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐷)))
18	14, 17	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑌 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐷)))
19	18	simprd 495	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))
20	18	simpld 494	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝐷))
21	3, 15, 10, 16, 19, 20	fucoppclem 49439	. . . 4 ⊢ (𝜑 → (𝑌𝑁𝑋) = ((𝐹‘𝑋)(𝑂 Nat 𝑃)(𝐹‘𝑌)))
22	14, 21	eleqtrd 2833	. . 3 ⊢ (𝜑 → 𝐴 ∈ ((𝐹‘𝑋)(𝑂 Nat 𝑃)(𝐹‘𝑌)))
23		fucoppcco.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝑌(Hom ‘𝑅)𝑍))
24	11, 12	oppchom 17616	. . . . 5 ⊢ (𝑌(Hom ‘𝑅)𝑍) = (𝑍𝑁𝑌)
25	23, 24	eleqtrdi 2841	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑍𝑁𝑌))
26	10	natrcl 17855	. . . . . . 7 ⊢ (𝐵 ∈ (𝑍𝑁𝑌) → (𝑍 ∈ (𝐶 Func 𝐷) ∧ 𝑌 ∈ (𝐶 Func 𝐷)))
27	25, 26	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑍 ∈ (𝐶 Func 𝐷) ∧ 𝑌 ∈ (𝐶 Func 𝐷)))
28	27	simpld 494	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝐶 Func 𝐷))
29	3, 15, 10, 16, 20, 28	fucoppclem 49439	. . . 4 ⊢ (𝜑 → (𝑍𝑁𝑌) = ((𝐹‘𝑌)(𝑂 Nat 𝑃)(𝐹‘𝑍)))
30	25, 29	eleqtrd 2833	. . 3 ⊢ (𝜑 → 𝐵 ∈ ((𝐹‘𝑌)(𝑂 Nat 𝑃)(𝐹‘𝑍)))
31	1, 2, 5, 6, 7, 22, 30	fucco 17867	. 2 ⊢ (𝜑 → (𝐵(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))𝐴) = (𝑧 ∈ (Base‘𝐶) ↦ ((𝐵‘𝑧)(⟨((1^st ‘(𝐹‘𝑋))‘𝑧), ((1^st ‘(𝐹‘𝑌))‘𝑧)⟩(comp‘𝑃)((1^st ‘(𝐹‘𝑍))‘𝑧))(𝐴‘𝑧))))
32		fucoppc.g	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))
33		eqidd 2732	. . . 4 ⊢ (𝜑 → 𝐵 = 𝐵)
34	32, 20, 28, 33, 25	opf2 49438	. . 3 ⊢ (𝜑 → ((𝑌𝐺𝑍)‘𝐵) = 𝐵)
35		eqidd 2732	. . . 4 ⊢ (𝜑 → 𝐴 = 𝐴)
36	32, 19, 20, 35, 14	opf2 49438	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝐴) = 𝐴)
37	34, 36	oveq12d 7359	. 2 ⊢ (𝜑 → (((𝑌𝐺𝑍)‘𝐵)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐴)) = (𝐵(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))𝐴))
38		eqid 2731	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
39		eqid 2731	. . . 4 ⊢ (comp‘𝑄) = (comp‘𝑄)
40	9, 10, 4, 38, 39, 25, 14	fucco 17867	. . 3 ⊢ (𝜑 → (𝐴(⟨𝑍, 𝑌⟩(comp‘𝑄)𝑋)𝐵) = (𝑧 ∈ (Base‘𝐶) ↦ ((𝐴‘𝑧)(⟨((1^st ‘𝑍)‘𝑧), ((1^st ‘𝑌)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝑋)‘𝑧))(𝐵‘𝑧))))
41	9	fucbas 17865	. . . . 5 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
42	41, 39, 12, 19, 20, 28	oppcco 17618	. . . 4 ⊢ (𝜑 → (𝐵(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐴) = (𝐴(⟨𝑍, 𝑌⟩(comp‘𝑄)𝑋)𝐵))
43	9, 10, 39, 25, 14	fuccocl 17869	. . . 4 ⊢ (𝜑 → (𝐴(⟨𝑍, 𝑌⟩(comp‘𝑄)𝑋)𝐵) ∈ (𝑍𝑁𝑋))
44	32, 19, 28, 42, 43	opf2 49438	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑍)‘(𝐵(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐴)) = (𝐴(⟨𝑍, 𝑌⟩(comp‘𝑄)𝑋)𝐵))
45	16, 19	opf11 49435	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘(𝐹‘𝑋)) = (1^st ‘𝑋))
46	45	fveq1d 6819	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘(𝐹‘𝑋))‘𝑧) = ((1^st ‘𝑋)‘𝑧))
47	16, 20	opf11 49435	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘(𝐹‘𝑌)) = (1^st ‘𝑌))
48	47	fveq1d 6819	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘(𝐹‘𝑌))‘𝑧) = ((1^st ‘𝑌)‘𝑧))
49	46, 48	opeq12d 4828	. . . . . . . 8 ⊢ (𝜑 → ⟨((1^st ‘(𝐹‘𝑋))‘𝑧), ((1^st ‘(𝐹‘𝑌))‘𝑧)⟩ = ⟨((1^st ‘𝑋)‘𝑧), ((1^st ‘𝑌)‘𝑧)⟩)
50	16, 28	opf11 49435	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘(𝐹‘𝑍)) = (1^st ‘𝑍))
51	50	fveq1d 6819	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘(𝐹‘𝑍))‘𝑧) = ((1^st ‘𝑍)‘𝑧))
52	49, 51	oveq12d 7359	. . . . . . 7 ⊢ (𝜑 → (⟨((1^st ‘(𝐹‘𝑋))‘𝑧), ((1^st ‘(𝐹‘𝑌))‘𝑧)⟩(comp‘𝑃)((1^st ‘(𝐹‘𝑍))‘𝑧)) = (⟨((1^st ‘𝑋)‘𝑧), ((1^st ‘𝑌)‘𝑧)⟩(comp‘𝑃)((1^st ‘𝑍)‘𝑧)))
53	52	oveqd 7358	. . . . . 6 ⊢ (𝜑 → ((𝐵‘𝑧)(⟨((1^st ‘(𝐹‘𝑋))‘𝑧), ((1^st ‘(𝐹‘𝑌))‘𝑧)⟩(comp‘𝑃)((1^st ‘(𝐹‘𝑍))‘𝑧))(𝐴‘𝑧)) = ((𝐵‘𝑧)(⟨((1^st ‘𝑋)‘𝑧), ((1^st ‘𝑌)‘𝑧)⟩(comp‘𝑃)((1^st ‘𝑍)‘𝑧))(𝐴‘𝑧)))
54	53	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐶)) → ((𝐵‘𝑧)(⟨((1^st ‘(𝐹‘𝑋))‘𝑧), ((1^st ‘(𝐹‘𝑌))‘𝑧)⟩(comp‘𝑃)((1^st ‘(𝐹‘𝑍))‘𝑧))(𝐴‘𝑧)) = ((𝐵‘𝑧)(⟨((1^st ‘𝑋)‘𝑧), ((1^st ‘𝑌)‘𝑧)⟩(comp‘𝑃)((1^st ‘𝑍)‘𝑧))(𝐴‘𝑧)))
55		eqid 2731	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
56	19	func1st2nd 49108	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑋)(𝐶 Func 𝐷)(2^nd ‘𝑋))
57	4, 55, 56	funcf1 17768	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑋):(Base‘𝐶)⟶(Base‘𝐷))
58	57	ffvelcdmda 7012	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐶)) → ((1^st ‘𝑋)‘𝑧) ∈ (Base‘𝐷))
59	20	func1st2nd 49108	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑌)(𝐶 Func 𝐷)(2^nd ‘𝑌))
60	4, 55, 59	funcf1 17768	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑌):(Base‘𝐶)⟶(Base‘𝐷))
61	60	ffvelcdmda 7012	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐶)) → ((1^st ‘𝑌)‘𝑧) ∈ (Base‘𝐷))
62	28	func1st2nd 49108	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑍)(𝐶 Func 𝐷)(2^nd ‘𝑍))
63	4, 55, 62	funcf1 17768	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑍):(Base‘𝐶)⟶(Base‘𝐷))
64	63	ffvelcdmda 7012	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐶)) → ((1^st ‘𝑍)‘𝑧) ∈ (Base‘𝐷))
65	55, 38, 15, 58, 61, 64	oppcco 17618	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐶)) → ((𝐵‘𝑧)(⟨((1^st ‘𝑋)‘𝑧), ((1^st ‘𝑌)‘𝑧)⟩(comp‘𝑃)((1^st ‘𝑍)‘𝑧))(𝐴‘𝑧)) = ((𝐴‘𝑧)(⟨((1^st ‘𝑍)‘𝑧), ((1^st ‘𝑌)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝑋)‘𝑧))(𝐵‘𝑧)))
66	54, 65	eqtrd 2766	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐶)) → ((𝐵‘𝑧)(⟨((1^st ‘(𝐹‘𝑋))‘𝑧), ((1^st ‘(𝐹‘𝑌))‘𝑧)⟩(comp‘𝑃)((1^st ‘(𝐹‘𝑍))‘𝑧))(𝐴‘𝑧)) = ((𝐴‘𝑧)(⟨((1^st ‘𝑍)‘𝑧), ((1^st ‘𝑌)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝑋)‘𝑧))(𝐵‘𝑧)))
67	66	mpteq2dva 5179	. . 3 ⊢ (𝜑 → (𝑧 ∈ (Base‘𝐶) ↦ ((𝐵‘𝑧)(⟨((1^st ‘(𝐹‘𝑋))‘𝑧), ((1^st ‘(𝐹‘𝑌))‘𝑧)⟩(comp‘𝑃)((1^st ‘(𝐹‘𝑍))‘𝑧))(𝐴‘𝑧))) = (𝑧 ∈ (Base‘𝐶) ↦ ((𝐴‘𝑧)(⟨((1^st ‘𝑍)‘𝑧), ((1^st ‘𝑌)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝑋)‘𝑧))(𝐵‘𝑧))))
68	40, 44, 67	3eqtr4d 2776	. 2 ⊢ (𝜑 → ((𝑋𝐺𝑍)‘(𝐵(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐴)) = (𝑧 ∈ (Base‘𝐶) ↦ ((𝐵‘𝑧)(⟨((1^st ‘(𝐹‘𝑋))‘𝑧), ((1^st ‘(𝐹‘𝑌))‘𝑧)⟩(comp‘𝑃)((1^st ‘(𝐹‘𝑍))‘𝑧))(𝐴‘𝑧))))
69	31, 37, 68	3eqtr4rd 2777	1 ⊢ (𝜑 → ((𝑋𝐺𝑍)‘(𝐵(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐴)) = (((𝑌𝐺𝑍)‘𝐵)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ⟨cop 4577 ↦ cmpt 5167 I cid 5505 ↾ cres 5613 ‘cfv 6476 (class class class)co 7341 ∈ cmpo 7343 1^st c1st 7914 2^nd c2nd 7915 Basecbs 17115 Hom chom 17167 compcco 17168 oppCatcoppc 17612 Func cfunc 17756 Nat cnat 17846 FuncCat cfuc 17847 oppFunc coppf 49154

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 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 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-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-tpos 8151 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-er 8617 df-map 8747 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-z 12464 df-dec 12584 df-uz 12728 df-fz 13403 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-hom 17180 df-cco 17181 df-cat 17569 df-cid 17570 df-homf 17571 df-comf 17572 df-oppc 17613 df-func 17760 df-nat 17848 df-fuc 17849 df-oppf 49155

This theorem is referenced by: fucoppc 49442

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