fucoco2 - Mathbox for Zhi Wang

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Theorem fucoco2 49706

Description: Composition in the source category is mapped to composition in the target. See also fucoco 49705. (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 49600	. . . . 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 2839	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
8		fucoco2.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑊)
9	8, 6	eleqtrd 2839	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
10	2, 3, 4, 7, 9	xpcfuchom 49602	. . . 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 2839	. . 3 ⊢ (𝜑 → 𝐴 ∈ (((1^st ‘𝑋)(𝐷 Nat 𝐸)(1^st ‘𝑌)) × ((2^nd ‘𝑋)(𝐶 Nat 𝐷)(2^nd ‘𝑌))))
12		xp1st 7975	. . 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 7976	. . 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 2839	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
19	2, 3, 4, 9, 18	xpcfuchom 49602	. . . 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 2839	. . 3 ⊢ (𝜑 → 𝐵 ∈ (((1^st ‘𝑌)(𝐷 Nat 𝐸)(1^st ‘𝑍)) × ((2^nd ‘𝑌)(𝐶 Nat 𝐷)(2^nd ‘𝑍))))
21		xp1st 7975	. . 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 7976	. . 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 7982	. . 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 7982	. . 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 7982	. . 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 7982	. . 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 7982	. . 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 49705	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 1542 ∈ wcel 2114 ⟨cop 4588 × cxp 5630 ‘cfv 6500 (class class class)co 7368 1^st c1st 7941 2^nd c2nd 7942 Hom chom 17200 compcco 17201 Func cfunc 17790 Nat cnat 17880 FuncCat cfuc 17881 ×_c cxpc 18103 ∘_F cfuco 49664

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17149 df-hom 17213 df-cco 17214 df-cat 17603 df-cid 17604 df-func 17794 df-cofu 17796 df-nat 17882 df-fuc 17883 df-xpc 18107 df-fuco 49665

This theorem is referenced by: fucofunc 49707

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