cofuoppf - Mathbox for Zhi Wang

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Theorem cofuoppf 49145

Description: Composition of opposite functors. (Contributed by Zhi Wang, 26-Nov-2025.)

**Hypotheses**
Ref	Expression
cofuoppf.k	⊢ (𝜑 → (𝐺 ∘_func 𝐹) = 𝐾)
cofuoppf.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
cofuoppf.g	⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))

**Assertion**
Ref	Expression
cofuoppf	⊢ (𝜑 → (( oppFunc ‘𝐺) ∘_func ( oppFunc ‘𝐹)) = ( oppFunc ‘𝐾))

Proof of Theorem cofuoppf Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . . 4 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
2		eqid 2729	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	oppcbas 17624	. . 3 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝐶))
4		eqid 2729	. . . 4 ⊢ (oppCat‘𝐷) = (oppCat‘𝐷)
5		cofuoppf.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
6	5	func1st2nd 49071	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
7	1, 4, 6	funcoppc 17782	. . 3 ⊢ (𝜑 → (1^st ‘𝐹)((oppCat‘𝐶) Func (oppCat‘𝐷))tpos (2^nd ‘𝐹))
8		eqid 2729	. . . 4 ⊢ (oppCat‘𝐸) = (oppCat‘𝐸)
9		cofuoppf.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
10	9	func1st2nd 49071	. . . 4 ⊢ (𝜑 → (1^st ‘𝐺)(𝐷 Func 𝐸)(2^nd ‘𝐺))
11	4, 8, 10	funcoppc 17782	. . 3 ⊢ (𝜑 → (1^st ‘𝐺)((oppCat‘𝐷) Func (oppCat‘𝐸))tpos (2^nd ‘𝐺))
12	3, 7, 11	cofuval2 17794	. 2 ⊢ (𝜑 → (⟨(1^st ‘𝐺), tpos (2^nd ‘𝐺)⟩ ∘_func ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩) = ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), (𝑦 ∈ (Base‘𝐶), 𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑦)tpos (2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥)) ∘ (𝑦tpos (2^nd ‘𝐹)𝑥)))⟩)
13		oppfval2 49132	. . . 4 ⊢ (𝐺 ∈ (𝐷 Func 𝐸) → ( oppFunc ‘𝐺) = ⟨(1^st ‘𝐺), tpos (2^nd ‘𝐺)⟩)
14	9, 13	syl 17	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐺) = ⟨(1^st ‘𝐺), tpos (2^nd ‘𝐺)⟩)
15		oppfval2 49132	. . . 4 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
16	5, 15	syl 17	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
17	14, 16	oveq12d 7367	. 2 ⊢ (𝜑 → (( oppFunc ‘𝐺) ∘_func ( oppFunc ‘𝐹)) = (⟨(1^st ‘𝐺), tpos (2^nd ‘𝐺)⟩ ∘_func ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩))
18		cofuoppf.k	. . . . 5 ⊢ (𝜑 → (𝐺 ∘_func 𝐹) = 𝐾)
19	2, 5, 9	cofuval 17789	. . . . 5 ⊢ (𝜑 → (𝐺 ∘_func 𝐹) = ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦)))⟩)
20	18, 19	eqtr3d 2766	. . . 4 ⊢ (𝜑 → 𝐾 = ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦)))⟩)
21	5, 9	cofucl 17795	. . . . 5 ⊢ (𝜑 → (𝐺 ∘_func 𝐹) ∈ (𝐶 Func 𝐸))
22	18, 21	eqeltrrd 2829	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝐶 Func 𝐸))
23	20, 22	oppfval3 49133	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐾) = ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), tpos (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦)))⟩)
24		ovtpos 8174	. . . . . . . . 9 ⊢ (((1^st ‘𝐹)‘𝑦)tpos (2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥)) = (((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦))
25		ovtpos 8174	. . . . . . . . 9 ⊢ (𝑦tpos (2^nd ‘𝐹)𝑥) = (𝑥(2^nd ‘𝐹)𝑦)
26	24, 25	coeq12i 5806	. . . . . . . 8 ⊢ ((((1^st ‘𝐹)‘𝑦)tpos (2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥)) ∘ (𝑦tpos (2^nd ‘𝐹)𝑥)) = ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦))
27	26	eqcomi 2738	. . . . . . 7 ⊢ ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦)) = ((((1^st ‘𝐹)‘𝑦)tpos (2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥)) ∘ (𝑦tpos (2^nd ‘𝐹)𝑥))
28	27	a1i 11	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦)) = ((((1^st ‘𝐹)‘𝑦)tpos (2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥)) ∘ (𝑦tpos (2^nd ‘𝐹)𝑥)))
29	28	mpoeq3ia 7427	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑦)tpos (2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥)) ∘ (𝑦tpos (2^nd ‘𝐹)𝑥)))
30	29	tposmpo 8196	. . . 4 ⊢ tpos (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦))) = (𝑦 ∈ (Base‘𝐶), 𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑦)tpos (2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥)) ∘ (𝑦tpos (2^nd ‘𝐹)𝑥)))
31	30	opeq2i 4828	. . 3 ⊢ ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), tpos (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦)))⟩ = ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), (𝑦 ∈ (Base‘𝐶), 𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑦)tpos (2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥)) ∘ (𝑦tpos (2^nd ‘𝐹)𝑥)))⟩
32	23, 31	eqtrdi 2780	. 2 ⊢ (𝜑 → ( oppFunc ‘𝐾) = ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), (𝑦 ∈ (Base‘𝐶), 𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐹)‘𝑦)tpos (2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥)) ∘ (𝑦tpos (2^nd ‘𝐹)𝑥)))⟩)
33	12, 17, 32	3eqtr4d 2774	1 ⊢ (𝜑 → (( oppFunc ‘𝐺) ∘_func ( oppFunc ‘𝐹)) = ( oppFunc ‘𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ⟨cop 4583 ∘ ccom 5623 ‘cfv 6482 (class class class)co 7349 ∈ cmpo 7351 1^st c1st 7922 2^nd c2nd 7923 tpos ctpos 8158 Basecbs 17120 oppCatcoppc 17617 Func cfunc 17761 ∘_func ccofu 17763 oppFunc coppf 49117

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 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

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 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-tpos 8159 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-er 8625 df-map 8755 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-hom 17185 df-cco 17186 df-cat 17574 df-cid 17575 df-oppc 17618 df-func 17765 df-cofu 17767 df-oppf 49118

This theorem is referenced by: lmdran 49666

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