fucoppcco - Mathbox for Zhi Wang

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Theorem fucoppcco 49896

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 2737	. . 3 ⊢ (𝑂 Nat 𝑃) = (𝑂 Nat 𝑃)
3		fucoppc.o	. . . 4 ⊢ 𝑂 = (oppCat‘𝐶)
4		eqid 2737	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	3, 4	oppcbas 17675	. . 3 ⊢ (Base‘𝐶) = (Base‘𝑂)
6		eqid 2737	. . 3 ⊢ (comp‘𝑃) = (comp‘𝑃)
7		eqid 2737	. . 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 17922	. . . . . 6 ⊢ 𝑁 = (Hom ‘𝑄)
12		fucoppc.r	. . . . . 6 ⊢ 𝑅 = (oppCat‘𝑄)
13	11, 12	oppchom 17672	. . . . 5 ⊢ (𝑋(Hom ‘𝑅)𝑌) = (𝑌𝑁𝑋)
14	8, 13	eleqtrdi 2847	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝑌𝑁𝑋))
15		fucoppc.p	. . . . 5 ⊢ 𝑃 = (oppCat‘𝐷)
16		fucoppc.f	. . . . 5 ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
17	10	natrcl 17911	. . . . . . 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 49894	. . . 4 ⊢ (𝜑 → (𝑌𝑁𝑋) = ((𝐹‘𝑋)(𝑂 Nat 𝑃)(𝐹‘𝑌)))
22	14, 21	eleqtrd 2839	. . 3 ⊢ (𝜑 → 𝐴 ∈ ((𝐹‘𝑋)(𝑂 Nat 𝑃)(𝐹‘𝑌)))
23		fucoppcco.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝑌(Hom ‘𝑅)𝑍))
24	11, 12	oppchom 17672	. . . . 5 ⊢ (𝑌(Hom ‘𝑅)𝑍) = (𝑍𝑁𝑌)
25	23, 24	eleqtrdi 2847	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑍𝑁𝑌))
26	10	natrcl 17911	. . . . . . 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 49894	. . . 4 ⊢ (𝜑 → (𝑍𝑁𝑌) = ((𝐹‘𝑌)(𝑂 Nat 𝑃)(𝐹‘𝑍)))
30	25, 29	eleqtrd 2839	. . 3 ⊢ (𝜑 → 𝐵 ∈ ((𝐹‘𝑌)(𝑂 Nat 𝑃)(𝐹‘𝑍)))
31	1, 2, 5, 6, 7, 22, 30	fucco 17923	. 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 2738	. . . 4 ⊢ (𝜑 → 𝐵 = 𝐵)
34	32, 20, 28, 33, 25	opf2 49893	. . 3 ⊢ (𝜑 → ((𝑌𝐺𝑍)‘𝐵) = 𝐵)
35		eqidd 2738	. . . 4 ⊢ (𝜑 → 𝐴 = 𝐴)
36	32, 19, 20, 35, 14	opf2 49893	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝐴) = 𝐴)
37	34, 36	oveq12d 7378	. 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 2737	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
39		eqid 2737	. . . 4 ⊢ (comp‘𝑄) = (comp‘𝑄)
40	9, 10, 4, 38, 39, 25, 14	fucco 17923	. . 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 17921	. . . . 5 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
42	41, 39, 12, 19, 20, 28	oppcco 17674	. . . 4 ⊢ (𝜑 → (𝐵(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐴) = (𝐴(⟨𝑍, 𝑌⟩(comp‘𝑄)𝑋)𝐵))
43	9, 10, 39, 25, 14	fuccocl 17925	. . . 4 ⊢ (𝜑 → (𝐴(⟨𝑍, 𝑌⟩(comp‘𝑄)𝑋)𝐵) ∈ (𝑍𝑁𝑋))
44	32, 19, 28, 42, 43	opf2 49893	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑍)‘(𝐵(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐴)) = (𝐴(⟨𝑍, 𝑌⟩(comp‘𝑄)𝑋)𝐵))
45	16, 19	opf11 49890	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘(𝐹‘𝑋)) = (1^st ‘𝑋))
46	45	fveq1d 6836	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘(𝐹‘𝑋))‘𝑧) = ((1^st ‘𝑋)‘𝑧))
47	16, 20	opf11 49890	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘(𝐹‘𝑌)) = (1^st ‘𝑌))
48	47	fveq1d 6836	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘(𝐹‘𝑌))‘𝑧) = ((1^st ‘𝑌)‘𝑧))
49	46, 48	opeq12d 4825	. . . . . . . 8 ⊢ (𝜑 → ⟨((1^st ‘(𝐹‘𝑋))‘𝑧), ((1^st ‘(𝐹‘𝑌))‘𝑧)⟩ = ⟨((1^st ‘𝑋)‘𝑧), ((1^st ‘𝑌)‘𝑧)⟩)
50	16, 28	opf11 49890	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘(𝐹‘𝑍)) = (1^st ‘𝑍))
51	50	fveq1d 6836	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘(𝐹‘𝑍))‘𝑧) = ((1^st ‘𝑍)‘𝑧))
52	49, 51	oveq12d 7378	. . . . . . 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 7377	. . . . . 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 2737	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
56	19	func1st2nd 49563	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑋)(𝐶 Func 𝐷)(2^nd ‘𝑋))
57	4, 55, 56	funcf1 17824	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑋):(Base‘𝐶)⟶(Base‘𝐷))
58	57	ffvelcdmda 7030	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐶)) → ((1^st ‘𝑋)‘𝑧) ∈ (Base‘𝐷))
59	20	func1st2nd 49563	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑌)(𝐶 Func 𝐷)(2^nd ‘𝑌))
60	4, 55, 59	funcf1 17824	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑌):(Base‘𝐶)⟶(Base‘𝐷))
61	60	ffvelcdmda 7030	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐶)) → ((1^st ‘𝑌)‘𝑧) ∈ (Base‘𝐷))
62	28	func1st2nd 49563	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑍)(𝐶 Func 𝐷)(2^nd ‘𝑍))
63	4, 55, 62	funcf1 17824	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑍):(Base‘𝐶)⟶(Base‘𝐷))
64	63	ffvelcdmda 7030	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐶)) → ((1^st ‘𝑍)‘𝑧) ∈ (Base‘𝐷))
65	55, 38, 15, 58, 61, 64	oppcco 17674	. . . . 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 2772	. . . 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 2782	. 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 2783	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 1542 ∈ wcel 2114 ⟨cop 4574 ↦ cmpt 5167 I cid 5518 ↾ cres 5626 ‘cfv 6492 (class class class)co 7360 ∈ cmpo 7362 1^st c1st 7933 2^nd c2nd 7934 Basecbs 17170 Hom chom 17222 compcco 17223 oppCatcoppc 17668 Func cfunc 17812 Nat cnat 17902 FuncCat cfuc 17903 oppFunc coppf 49609

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-tpos 8169 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-map 8768 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-fz 13453 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-hom 17235 df-cco 17236 df-cat 17625 df-cid 17626 df-homf 17627 df-comf 17628 df-oppc 17669 df-func 17816 df-nat 17904 df-fuc 17905 df-oppf 49610

This theorem is referenced by: fucoppc 49897

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