fucoco2 - Mathbox for Zhi Wang

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Theorem fucoco2 49347

Description: Composition in the source category is mapped to composition in the target. See also fucoco 49346. (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 49241	. . . . 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 2830	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
8		fucoco2.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑊)
9	8, 6	eleqtrd 2830	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
10	2, 3, 4, 7, 9	xpcfuchom 49243	. . . 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 2830	. . 3 ⊢ (𝜑 → 𝐴 ∈ (((1^st ‘𝑋)(𝐷 Nat 𝐸)(1^st ‘𝑌)) × ((2^nd ‘𝑋)(𝐶 Nat 𝐷)(2^nd ‘𝑌))))
12		xp1st 8000	. . 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 8001	. . 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 2830	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
19	2, 3, 4, 9, 18	xpcfuchom 49243	. . . 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 2830	. . 3 ⊢ (𝜑 → 𝐵 ∈ (((1^st ‘𝑌)(𝐷 Nat 𝐸)(1^st ‘𝑍)) × ((2^nd ‘𝑌)(𝐶 Nat 𝐷)(2^nd ‘𝑍))))
21		xp1st 8000	. . 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 8001	. . 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 8007	. . 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 8007	. . 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 8007	. . 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 8007	. . 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 8007	. . 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 49346	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 1540 ∈ wcel 2109 ⟨cop 4595 × cxp 5636 ‘cfv 6511 (class class class)co 7387 1^st c1st 7966 2^nd c2nd 7967 Hom chom 17231 compcco 17232 Func cfunc 17816 Nat cnat 17906 FuncCat cfuc 17907 ×_c cxpc 18129 ∘_F cfuco 49305

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 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

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 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-fz 13469 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-hom 17244 df-cco 17245 df-cat 17629 df-cid 17630 df-func 17820 df-cofu 17822 df-nat 17908 df-fuc 17909 df-xpc 18133 df-fuco 49306

This theorem is referenced by: fucofunc 49348

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