cofuoppf - Mathbox for Zhi Wang

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Theorem cofuoppf 50469

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 2763	. . . 4 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
2		eqid 2763	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	oppcbas 17918	. . 3 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝐶))
4		eqid 2763	. . . 4 ⊢ (oppCat‘𝐷) = (oppCat‘𝐷)
5		cofuoppf.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
6	5	func1st2nd 50393	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
7	1, 4, 6	funcoppc 18076	. . 3 ⊢ (𝜑 → (1^st ‘𝐹)((oppCat‘𝐶) Func (oppCat‘𝐷))tpos (2^nd ‘𝐹))
8		eqid 2763	. . . 4 ⊢ (oppCat‘𝐸) = (oppCat‘𝐸)
9		cofuoppf.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
10	9	func1st2nd 50393	. . . 4 ⊢ (𝜑 → (1^st ‘𝐺)(𝐷 Func 𝐸)(2^nd ‘𝐺))
11	4, 8, 10	funcoppc 18076	. . 3 ⊢ (𝜑 → (1^st ‘𝐺)((oppCat‘𝐷) Func (oppCat‘𝐸))tpos (2^nd ‘𝐺))
12	3, 7, 11	cofuval2 18088	. 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 50456	. . . 4 ⊢ (𝐺 ∈ (𝐷 Func 𝐸) → ( oppFunc ‘𝐺) = ⟨(1^st ‘𝐺), tpos (2^nd ‘𝐺)⟩)
14	9, 13	syl 18	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐺) = ⟨(1^st ‘𝐺), tpos (2^nd ‘𝐺)⟩)
15		oppfval2 50456	. . . 4 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
16	5, 15	syl 18	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
17	14, 16	oveq12d 7458	. 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 18083	. . . . 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 2800	. . . 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 18089	. . . . 5 ⊢ (𝜑 → (𝐺 ∘_func 𝐹) ∈ (𝐶 Func 𝐸))
22	18, 21	eqeltrrd 2865	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝐶 Func 𝐸))
23	20, 22	oppfval3 50457	. . 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 8283	. . . . . . . . 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 8283	. . . . . . . . 9 ⊢ (𝑦tpos (2^nd ‘𝐹)𝑥) = (𝑥(2^nd ‘𝐹)𝑦)
26	24, 25	coeq12i 5846	. . . . . . . 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 2772	. . . . . . 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 7518	. . . . 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 8305	. . . 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 4847	. . 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 2814	. 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 2808	1 ⊢ (𝜑 → (( oppFunc ‘𝐺) ∘_func ( oppFunc ‘𝐹)) = ( oppFunc ‘𝐾))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ⟨cop 4838 ∘ ccom 5659 ‘cfv 6555 (class class class)co 7440 ∈ cmpo 7442 1^st c1st 8020 2^nd c2nd 8021 tpos ctpos 8267 Basecbs 17413 oppCatcoppc 17911 Func cfunc 18055 ∘_func ccofu 18057 oppFunc coppf 50441

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-om 7899 df-1st 8022 df-2nd 8023 df-tpos 8268 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-er 8742 df-map 8874 df-ixp 8951 df-en 8999 df-dom 9000 df-sdom 9001 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-z 12718 df-dec 12839 df-sets 17368 df-slot 17386 df-ndx 17398 df-base 17414 df-hom 17478 df-cco 17479 df-cat 17868 df-cid 17869 df-oppc 17912 df-func 18059 df-cofu 18061 df-oppf 50442

This theorem is used by: lmdran 51057

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