cofuoppf - Mathbox for Zhi Wang

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Theorem cofuoppf 50061

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 2762	. . . 4 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
2		eqid 2762	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	oppcbas 17808	. . 3 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝐶))
4		eqid 2762	. . . 4 ⊢ (oppCat‘𝐷) = (oppCat‘𝐷)
5		cofuoppf.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
6	5	func1st2nd 49987	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
7	1, 4, 6	funcoppc 17966	. . 3 ⊢ (𝜑 → (1^st ‘𝐹)((oppCat‘𝐶) Func (oppCat‘𝐷))tpos (2^nd ‘𝐹))
8		eqid 2762	. . . 4 ⊢ (oppCat‘𝐸) = (oppCat‘𝐸)
9		cofuoppf.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
10	9	func1st2nd 49987	. . . 4 ⊢ (𝜑 → (1^st ‘𝐺)(𝐷 Func 𝐸)(2^nd ‘𝐺))
11	4, 8, 10	funcoppc 17966	. . 3 ⊢ (𝜑 → (1^st ‘𝐺)((oppCat‘𝐷) Func (oppCat‘𝐸))tpos (2^nd ‘𝐺))
12	3, 7, 11	cofuval2 17978	. 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 50048	. . . 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 50048	. . . 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 7434	. 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 17973	. . . . 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 2799	. . . 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 17979	. . . . 5 ⊢ (𝜑 → (𝐺 ∘_func 𝐹) ∈ (𝐶 Func 𝐸))
22	18, 21	eqeltrrd 2863	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝐶 Func 𝐸))
23	20, 22	oppfval3 50049	. . 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 8242	. . . . . . . . 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 8242	. . . . . . . . 9 ⊢ (𝑦tpos (2^nd ‘𝐹)𝑥) = (𝑥(2^nd ‘𝐹)𝑦)
26	24, 25	coeq12i 5847	. . . . . . . 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 2771	. . . . . . 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 7494	. . . . 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 8264	. . . 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 4840	. . 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 2813	. 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 2807	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 2145 ⟨cop 4593 ∘ ccom 5663 ‘cfv 6537 (class class class)co 7416 ∈ cmpo 7418 1^st c1st 7987 2^nd c2nd 7988 tpos ctpos 8226 Basecbs 17303 oppCatcoppc 17801 Func cfunc 17945 ∘_func ccofu 17947 oppFunc coppf 50033

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202

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 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-tpos 8227 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-er 8699 df-map 8831 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-sets 17258 df-slot 17276 df-ndx 17288 df-base 17304 df-hom 17368 df-cco 17369 df-cat 17758 df-cid 17759 df-oppc 17802 df-func 17949 df-cofu 17951 df-oppf 50034

This theorem is used by: lmdran 50582

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