fucoco2 - Mathbox for Zhi Wang

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Theorem fucoco2 50164

Description: Composition in the source category is mapped to composition in the target. See also fucoco 50163. (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 50058	. . . . 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 2864	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
8		fucoco2.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑊)
9	8, 6	eleqtrd 2864	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
10	2, 3, 4, 7, 9	xpcfuchom 50060	. . . 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 2864	. . 3 ⊢ (𝜑 → 𝐴 ∈ (((1^st ‘𝑋)(𝐷 Nat 𝐸)(1^st ‘𝑌)) × ((2^nd ‘𝑋)(𝐶 Nat 𝐷)(2^nd ‘𝑌))))
12		xp1st 8016	. . 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 18	. 2 ⊢ (𝜑 → (1^st ‘𝐴) ∈ ((1^st ‘𝑋)(𝐷 Nat 𝐸)(1^st ‘𝑌)))
14		xp2nd 8017	. . 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 18	. 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 2864	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
19	2, 3, 4, 9, 18	xpcfuchom 50060	. . . 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 2864	. . 3 ⊢ (𝜑 → 𝐵 ∈ (((1^st ‘𝑌)(𝐷 Nat 𝐸)(1^st ‘𝑍)) × ((2^nd ‘𝑌)(𝐶 Nat 𝐷)(2^nd ‘𝑍))))
21		xp1st 8016	. . 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 18	. 2 ⊢ (𝜑 → (1^st ‘𝐵) ∈ ((1^st ‘𝑌)(𝐷 Nat 𝐸)(1^st ‘𝑍)))
23		xp2nd 8017	. . 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 18	. 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 8023	. . 3 ⊢ (𝑋 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
27	7, 26	syl 18	. 2 ⊢ (𝜑 → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
28		1st2nd2 8023	. . 3 ⊢ (𝑌 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) → 𝑌 = ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩)
29	9, 28	syl 18	. 2 ⊢ (𝜑 → 𝑌 = ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩)
30		1st2nd2 8023	. . 3 ⊢ (𝑍 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) → 𝑍 = ⟨(1^st ‘𝑍), (2^nd ‘𝑍)⟩)
31	18, 30	syl 18	. 2 ⊢ (𝜑 → 𝑍 = ⟨(1^st ‘𝑍), (2^nd ‘𝑍)⟩)
32		1st2nd2 8023	. . 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 18	. 2 ⊢ (𝜑 → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
34		1st2nd2 8023	. . 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 18	. 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 50163	1 ⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝐵(⟨𝑋, 𝑌⟩ · 𝑍)𝐴)) = (((𝑌𝑃𝑍)‘𝐵)(⟨(𝑂‘𝑋), (𝑂‘𝑌)⟩ ∙ (𝑂‘𝑍))((𝑋𝑃𝑌)‘𝐴)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1569 ∈ wcel 2142 ⟨cop 4594 × cxp 5658 ‘cfv 6536 (class class class)co 7412 1^st c1st 7982 2^nd c2nd 7983 Hom chom 17327 compcco 17328 Func cfunc 17917 Nat cnat 18007 FuncCat cfuc 18008 ×_c cxpc 18230 ∘_F cfuco 50122

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 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 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-er 8692 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-fz 13542 df-struct 17213 df-slot 17248 df-ndx 17260 df-base 17276 df-hom 17340 df-cco 17341 df-cat 17730 df-cid 17731 df-func 17921 df-cofu 17923 df-nat 18009 df-fuc 18010 df-xpc 18234 df-fuco 50123

This theorem is used by: fucofunc 50165

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