fucoco2 - Mathbox for Zhi Wang

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Theorem fucoco2 49473

Description: Composition in the source category is mapped to composition in the target. See also fucoco 49472. (Contributed by Zhi Wang, 3-Oct-2025.)

**Hypotheses**
Ref	Expression
fucoco2.t	⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
fucoco2.q	⊢ 𝑄 = (𝐶 FuncCat 𝐸)
fucoco2.o	⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
fucoco2.1	⊢ · = (comp‘𝑇)
fucoco2.2	⊢ ∙ = (comp‘𝑄)
fucoco2.w	⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
fucoco2.x	⊢ (𝜑 → 𝑋 ∈ 𝑊)
fucoco2.y	⊢ (𝜑 → 𝑌 ∈ 𝑊)
fucoco2.z	⊢ (𝜑 → 𝑍 ∈ 𝑊)
fucoco2.j	⊢ 𝐽 = (Hom ‘𝑇)
fucoco2.a	⊢ (𝜑 → 𝐴 ∈ (𝑋𝐽𝑌))
fucoco2.b	⊢ (𝜑 → 𝐵 ∈ (𝑌𝐽𝑍))

**Assertion**
Ref	Expression
fucoco2	⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝐵(⟨𝑋, 𝑌⟩ · 𝑍)𝐴)) = (((𝑌𝑃𝑍)‘𝐵)(⟨(𝑂‘𝑋), (𝑂‘𝑌)⟩ ∙ (𝑂‘𝑍))((𝑋𝑃𝑌)‘𝐴)))

**Proof of Theorem fucoco2**
Step	Hyp	Ref	Expression
1		fucoco2.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐽𝑌))
2		fucoco2.t	. . . . 5 ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
3	2	xpcfucbas 49367	. . . . 5 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = (Base‘𝑇)
4		fucoco2.j	. . . . 5 ⊢ 𝐽 = (Hom ‘𝑇)
5		fucoco2.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
6		fucoco2.w	. . . . . 6 ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
7	5, 6	eleqtrd 2835	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
8		fucoco2.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑊)
9	8, 6	eleqtrd 2835	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
10	2, 3, 4, 7, 9	xpcfuchom 49369	. . . 4 ⊢ (𝜑 → (𝑋𝐽𝑌) = (((1^st ‘𝑋)(𝐷 Nat 𝐸)(1^st ‘𝑌)) × ((2^nd ‘𝑋)(𝐶 Nat 𝐷)(2^nd ‘𝑌))))
11	1, 10	eleqtrd 2835	. . 3 ⊢ (𝜑 → 𝐴 ∈ (((1^st ‘𝑋)(𝐷 Nat 𝐸)(1^st ‘𝑌)) × ((2^nd ‘𝑋)(𝐶 Nat 𝐷)(2^nd ‘𝑌))))
12		xp1st 7962	. . 3 ⊢ (𝐴 ∈ (((1^st ‘𝑋)(𝐷 Nat 𝐸)(1^st ‘𝑌)) × ((2^nd ‘𝑋)(𝐶 Nat 𝐷)(2^nd ‘𝑌))) → (1^st ‘𝐴) ∈ ((1^st ‘𝑋)(𝐷 Nat 𝐸)(1^st ‘𝑌)))
13	11, 12	syl 17	. 2 ⊢ (𝜑 → (1^st ‘𝐴) ∈ ((1^st ‘𝑋)(𝐷 Nat 𝐸)(1^st ‘𝑌)))
14		xp2nd 7963	. . 3 ⊢ (𝐴 ∈ (((1^st ‘𝑋)(𝐷 Nat 𝐸)(1^st ‘𝑌)) × ((2^nd ‘𝑋)(𝐶 Nat 𝐷)(2^nd ‘𝑌))) → (2^nd ‘𝐴) ∈ ((2^nd ‘𝑋)(𝐶 Nat 𝐷)(2^nd ‘𝑌)))
15	11, 14	syl 17	. 2 ⊢ (𝜑 → (2^nd ‘𝐴) ∈ ((2^nd ‘𝑋)(𝐶 Nat 𝐷)(2^nd ‘𝑌)))
16		fucoco2.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑌𝐽𝑍))
17		fucoco2.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
18	17, 6	eleqtrd 2835	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
19	2, 3, 4, 9, 18	xpcfuchom 49369	. . . 4 ⊢ (𝜑 → (𝑌𝐽𝑍) = (((1^st ‘𝑌)(𝐷 Nat 𝐸)(1^st ‘𝑍)) × ((2^nd ‘𝑌)(𝐶 Nat 𝐷)(2^nd ‘𝑍))))
20	16, 19	eleqtrd 2835	. . 3 ⊢ (𝜑 → 𝐵 ∈ (((1^st ‘𝑌)(𝐷 Nat 𝐸)(1^st ‘𝑍)) × ((2^nd ‘𝑌)(𝐶 Nat 𝐷)(2^nd ‘𝑍))))
21		xp1st 7962	. . 3 ⊢ (𝐵 ∈ (((1^st ‘𝑌)(𝐷 Nat 𝐸)(1^st ‘𝑍)) × ((2^nd ‘𝑌)(𝐶 Nat 𝐷)(2^nd ‘𝑍))) → (1^st ‘𝐵) ∈ ((1^st ‘𝑌)(𝐷 Nat 𝐸)(1^st ‘𝑍)))
22	20, 21	syl 17	. 2 ⊢ (𝜑 → (1^st ‘𝐵) ∈ ((1^st ‘𝑌)(𝐷 Nat 𝐸)(1^st ‘𝑍)))
23		xp2nd 7963	. . 3 ⊢ (𝐵 ∈ (((1^st ‘𝑌)(𝐷 Nat 𝐸)(1^st ‘𝑍)) × ((2^nd ‘𝑌)(𝐶 Nat 𝐷)(2^nd ‘𝑍))) → (2^nd ‘𝐵) ∈ ((2^nd ‘𝑌)(𝐶 Nat 𝐷)(2^nd ‘𝑍)))
24	20, 23	syl 17	. 2 ⊢ (𝜑 → (2^nd ‘𝐵) ∈ ((2^nd ‘𝑌)(𝐶 Nat 𝐷)(2^nd ‘𝑍)))
25		fucoco2.o	. 2 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
26		1st2nd2 7969	. . 3 ⊢ (𝑋 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
27	7, 26	syl 17	. 2 ⊢ (𝜑 → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
28		1st2nd2 7969	. . 3 ⊢ (𝑌 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) → 𝑌 = ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩)
29	9, 28	syl 17	. 2 ⊢ (𝜑 → 𝑌 = ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩)
30		1st2nd2 7969	. . 3 ⊢ (𝑍 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) → 𝑍 = ⟨(1^st ‘𝑍), (2^nd ‘𝑍)⟩)
31	18, 30	syl 17	. 2 ⊢ (𝜑 → 𝑍 = ⟨(1^st ‘𝑍), (2^nd ‘𝑍)⟩)
32		1st2nd2 7969	. . 3 ⊢ (𝐴 ∈ (((1^st ‘𝑋)(𝐷 Nat 𝐸)(1^st ‘𝑌)) × ((2^nd ‘𝑋)(𝐶 Nat 𝐷)(2^nd ‘𝑌))) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
33	11, 32	syl 17	. 2 ⊢ (𝜑 → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
34		1st2nd2 7969	. . 3 ⊢ (𝐵 ∈ (((1^st ‘𝑌)(𝐷 Nat 𝐸)(1^st ‘𝑍)) × ((2^nd ‘𝑌)(𝐶 Nat 𝐷)(2^nd ‘𝑍))) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
35	20, 34	syl 17	. 2 ⊢ (𝜑 → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
36		fucoco2.q	. 2 ⊢ 𝑄 = (𝐶 FuncCat 𝐸)
37		fucoco2.2	. 2 ⊢ ∙ = (comp‘𝑄)
38		fucoco2.1	. 2 ⊢ · = (comp‘𝑇)
39	13, 15, 22, 24, 25, 27, 29, 31, 33, 35, 36, 37, 2, 38	fucoco 49472	1 ⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝐵(⟨𝑋, 𝑌⟩ · 𝑍)𝐴)) = (((𝑌𝑃𝑍)‘𝐵)(⟨(𝑂‘𝑋), (𝑂‘𝑌)⟩ ∙ (𝑂‘𝑍))((𝑋𝑃𝑌)‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 ⟨cop 4583 × cxp 5619 ‘cfv 6489 (class class class)co 7355 1^st c1st 7928 2^nd c2nd 7929 Hom chom 17182 compcco 17183 Func cfunc 17771 Nat cnat 17861 FuncCat cfuc 17862 ×_c cxpc 18084 ∘_F cfuco 49431

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11072 ax-resscn 11073 ax-1cn 11074 ax-icn 11075 ax-addcl 11076 ax-addrcl 11077 ax-mulcl 11078 ax-mulrcl 11079 ax-mulcom 11080 ax-addass 11081 ax-mulass 11082 ax-distr 11083 ax-i2m1 11084 ax-1ne0 11085 ax-1rid 11086 ax-rnegex 11087 ax-rrecex 11088 ax-cnre 11089 ax-pre-lttri 11090 ax-pre-lttrn 11091 ax-pre-ltadd 11092 ax-pre-mulgt0 11093

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-er 8631 df-map 8761 df-ixp 8831 df-en 8879 df-dom 8880 df-sdom 8881 df-fin 8882 df-pnf 11158 df-mnf 11159 df-xr 11160 df-ltxr 11161 df-le 11162 df-sub 11356 df-neg 11357 df-nn 12136 df-2 12198 df-3 12199 df-4 12200 df-5 12201 df-6 12202 df-7 12203 df-8 12204 df-9 12205 df-n0 12392 df-z 12479 df-dec 12599 df-uz 12743 df-fz 13418 df-struct 17068 df-slot 17103 df-ndx 17115 df-base 17131 df-hom 17195 df-cco 17196 df-cat 17584 df-cid 17585 df-func 17775 df-cofu 17777 df-nat 17863 df-fuc 17864 df-xpc 18088 df-fuco 49432

This theorem is referenced by: fucofunc 49474

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