fucoppcco - Mathbox for Zhi Wang

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Theorem fucoppcco 49414

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 2729	. . 3 ⊢ (𝑂 Nat 𝑃) = (𝑂 Nat 𝑃)
3		fucoppc.o	. . . 4 ⊢ 𝑂 = (oppCat‘𝐶)
4		eqid 2729	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	3, 4	oppcbas 17643	. . 3 ⊢ (Base‘𝐶) = (Base‘𝑂)
6		eqid 2729	. . 3 ⊢ (comp‘𝑃) = (comp‘𝑃)
7		eqid 2729	. . 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 17890	. . . . . 6 ⊢ 𝑁 = (Hom ‘𝑄)
12		fucoppc.r	. . . . . 6 ⊢ 𝑅 = (oppCat‘𝑄)
13	11, 12	oppchom 17640	. . . . 5 ⊢ (𝑋(Hom ‘𝑅)𝑌) = (𝑌𝑁𝑋)
14	8, 13	eleqtrdi 2838	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝑌𝑁𝑋))
15		fucoppc.p	. . . . 5 ⊢ 𝑃 = (oppCat‘𝐷)
16		fucoppc.f	. . . . 5 ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
17	10	natrcl 17879	. . . . . . 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 49412	. . . 4 ⊢ (𝜑 → (𝑌𝑁𝑋) = ((𝐹‘𝑋)(𝑂 Nat 𝑃)(𝐹‘𝑌)))
22	14, 21	eleqtrd 2830	. . 3 ⊢ (𝜑 → 𝐴 ∈ ((𝐹‘𝑋)(𝑂 Nat 𝑃)(𝐹‘𝑌)))
23		fucoppcco.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝑌(Hom ‘𝑅)𝑍))
24	11, 12	oppchom 17640	. . . . 5 ⊢ (𝑌(Hom ‘𝑅)𝑍) = (𝑍𝑁𝑌)
25	23, 24	eleqtrdi 2838	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑍𝑁𝑌))
26	10	natrcl 17879	. . . . . . 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 49412	. . . 4 ⊢ (𝜑 → (𝑍𝑁𝑌) = ((𝐹‘𝑌)(𝑂 Nat 𝑃)(𝐹‘𝑍)))
30	25, 29	eleqtrd 2830	. . 3 ⊢ (𝜑 → 𝐵 ∈ ((𝐹‘𝑌)(𝑂 Nat 𝑃)(𝐹‘𝑍)))
31	1, 2, 5, 6, 7, 22, 30	fucco 17891	. 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 2730	. . . 4 ⊢ (𝜑 → 𝐵 = 𝐵)
34	32, 20, 28, 33, 25	opf2 49411	. . 3 ⊢ (𝜑 → ((𝑌𝐺𝑍)‘𝐵) = 𝐵)
35		eqidd 2730	. . . 4 ⊢ (𝜑 → 𝐴 = 𝐴)
36	32, 19, 20, 35, 14	opf2 49411	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝐴) = 𝐴)
37	34, 36	oveq12d 7371	. 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 2729	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
39		eqid 2729	. . . 4 ⊢ (comp‘𝑄) = (comp‘𝑄)
40	9, 10, 4, 38, 39, 25, 14	fucco 17891	. . 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 17889	. . . . 5 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
42	41, 39, 12, 19, 20, 28	oppcco 17642	. . . 4 ⊢ (𝜑 → (𝐵(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐴) = (𝐴(⟨𝑍, 𝑌⟩(comp‘𝑄)𝑋)𝐵))
43	9, 10, 39, 25, 14	fuccocl 17893	. . . 4 ⊢ (𝜑 → (𝐴(⟨𝑍, 𝑌⟩(comp‘𝑄)𝑋)𝐵) ∈ (𝑍𝑁𝑋))
44	32, 19, 28, 42, 43	opf2 49411	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑍)‘(𝐵(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐴)) = (𝐴(⟨𝑍, 𝑌⟩(comp‘𝑄)𝑋)𝐵))
45	16, 19	opf11 49408	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘(𝐹‘𝑋)) = (1^st ‘𝑋))
46	45	fveq1d 6828	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘(𝐹‘𝑋))‘𝑧) = ((1^st ‘𝑋)‘𝑧))
47	16, 20	opf11 49408	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘(𝐹‘𝑌)) = (1^st ‘𝑌))
48	47	fveq1d 6828	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘(𝐹‘𝑌))‘𝑧) = ((1^st ‘𝑌)‘𝑧))
49	46, 48	opeq12d 4835	. . . . . . . 8 ⊢ (𝜑 → ⟨((1^st ‘(𝐹‘𝑋))‘𝑧), ((1^st ‘(𝐹‘𝑌))‘𝑧)⟩ = ⟨((1^st ‘𝑋)‘𝑧), ((1^st ‘𝑌)‘𝑧)⟩)
50	16, 28	opf11 49408	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘(𝐹‘𝑍)) = (1^st ‘𝑍))
51	50	fveq1d 6828	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘(𝐹‘𝑍))‘𝑧) = ((1^st ‘𝑍)‘𝑧))
52	49, 51	oveq12d 7371	. . . . . . 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 7370	. . . . . 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 2729	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
56	19	func1st2nd 49081	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑋)(𝐶 Func 𝐷)(2^nd ‘𝑋))
57	4, 55, 56	funcf1 17792	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑋):(Base‘𝐶)⟶(Base‘𝐷))
58	57	ffvelcdmda 7022	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐶)) → ((1^st ‘𝑋)‘𝑧) ∈ (Base‘𝐷))
59	20	func1st2nd 49081	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑌)(𝐶 Func 𝐷)(2^nd ‘𝑌))
60	4, 55, 59	funcf1 17792	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑌):(Base‘𝐶)⟶(Base‘𝐷))
61	60	ffvelcdmda 7022	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐶)) → ((1^st ‘𝑌)‘𝑧) ∈ (Base‘𝐷))
62	28	func1st2nd 49081	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑍)(𝐶 Func 𝐷)(2^nd ‘𝑍))
63	4, 55, 62	funcf1 17792	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑍):(Base‘𝐶)⟶(Base‘𝐷))
64	63	ffvelcdmda 7022	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐶)) → ((1^st ‘𝑍)‘𝑧) ∈ (Base‘𝐷))
65	55, 38, 15, 58, 61, 64	oppcco 17642	. . . . 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 2764	. . . 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 5188	. . 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 2774	. 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 2775	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 1540 ∈ wcel 2109 ⟨cop 4585 ↦ cmpt 5176 I cid 5517 ↾ cres 5625 ‘cfv 6486 (class class class)co 7353 ∈ cmpo 7355 1^st c1st 7929 2^nd c2nd 7930 Basecbs 17139 Hom chom 17191 compcco 17192 oppCatcoppc 17636 Func cfunc 17780 Nat cnat 17870 FuncCat cfuc 17871 oppFunc coppf 49127

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-map 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12611 df-uz 12755 df-fz 13430 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17140 df-hom 17204 df-cco 17205 df-cat 17593 df-cid 17594 df-homf 17595 df-comf 17596 df-oppc 17637 df-func 17784 df-nat 17872 df-fuc 17873 df-oppf 49128

This theorem is referenced by: fucoppc 49415

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