fucoco2 - Mathbox for Zhi Wang

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Theorem fucoco2 49353

Description: Composition in the source category is mapped to composition in the target. See also fucoco 49352. (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 49247	. . . . 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 49249	. . . 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 7956	. . 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 7957	. . 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 49249	. . . 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 7956	. . 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 7957	. . 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 7963	. . 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 7963	. . 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 7963	. . 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 7963	. . 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 7963	. . 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 49352	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 4583 × cxp 5617 ‘cfv 6482 (class class class)co 7349 1^st c1st 7922 2^nd c2nd 7923 Hom chom 17172 compcco 17173 Func cfunc 17761 Nat cnat 17851 FuncCat cfuc 17852 ×_c cxpc 18074 ∘_F cfuco 49311

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 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

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 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-er 8625 df-map 8755 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-fz 13411 df-struct 17058 df-slot 17093 df-ndx 17105 df-base 17121 df-hom 17185 df-cco 17186 df-cat 17574 df-cid 17575 df-func 17765 df-cofu 17767 df-nat 17853 df-fuc 17854 df-xpc 18078 df-fuco 49312

This theorem is referenced by: fucofunc 49354

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