cofuoppf - Mathbox for Zhi Wang

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Theorem cofuoppf 49129

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 2730	. . . 4 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
2		eqid 2730	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	oppcbas 17685	. . 3 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝐶))
4		eqid 2730	. . . 4 ⊢ (oppCat‘𝐷) = (oppCat‘𝐷)
5		cofuoppf.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
6	5	func1st2nd 49055	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
7	1, 4, 6	funcoppc 17843	. . 3 ⊢ (𝜑 → (1^st ‘𝐹)((oppCat‘𝐶) Func (oppCat‘𝐷))tpos (2^nd ‘𝐹))
8		eqid 2730	. . . 4 ⊢ (oppCat‘𝐸) = (oppCat‘𝐸)
9		cofuoppf.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
10	9	func1st2nd 49055	. . . 4 ⊢ (𝜑 → (1^st ‘𝐺)(𝐷 Func 𝐸)(2^nd ‘𝐺))
11	4, 8, 10	funcoppc 17843	. . 3 ⊢ (𝜑 → (1^st ‘𝐺)((oppCat‘𝐷) Func (oppCat‘𝐸))tpos (2^nd ‘𝐺))
12	3, 7, 11	cofuval2 17855	. 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 49116	. . . 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 49116	. . . 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 7407	. 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 17850	. . . . 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 2767	. . . 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 17856	. . . . 5 ⊢ (𝜑 → (𝐺 ∘_func 𝐹) ∈ (𝐶 Func 𝐸))
22	18, 21	eqeltrrd 2830	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝐶 Func 𝐸))
23	20, 22	oppfval3 49117	. . 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 8222	. . . . . . . . 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 8222	. . . . . . . . 9 ⊢ (𝑦tpos (2^nd ‘𝐹)𝑥) = (𝑥(2^nd ‘𝐹)𝑦)
26	24, 25	coeq12i 5829	. . . . . . . 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 2739	. . . . . . 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 7469	. . . . 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 8244	. . . 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 4843	. . 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 2781	. 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 2775	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 4597 ∘ ccom 5644 ‘cfv 6513 (class class class)co 7389 ∈ cmpo 7391 1^st c1st 7968 2^nd c2nd 7969 tpos ctpos 8206 Basecbs 17185 oppCatcoppc 17678 Func cfunc 17822 ∘_func ccofu 17824 oppFunc coppf 49101

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 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-1st 7970 df-2nd 7971 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-er 8673 df-map 8803 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-z 12536 df-dec 12656 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17186 df-hom 17250 df-cco 17251 df-cat 17635 df-cid 17636 df-oppc 17679 df-func 17826 df-cofu 17828 df-oppf 49102

This theorem is referenced by: lmdran 49650

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