fucofunc - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fucofunc		Structured version Visualization version GIF version

Theorem fucofunc 49988

Description: The functor composition bifunctor is a functor. See also fucofunca 49989.

However, it is unlikely the unique functor compatible with the functor composition. As a counterexample, let 𝐶 and 𝐷 be terminal categories (categories of one object and one morphism, df-termc 50102), for example, (SetCat‘1_o) (the trivial category, setc1oterm 50120), and 𝐸 be a category with two objects equipped with only two non-identity morphisms 𝑓 and 𝑔, pointing in the same direction. It is possible to map the ordered pair of natural transformations ⟨𝑎, 𝑖⟩, where 𝑎 sends to 𝑓 and 𝑖 is the identity natural transformation, to the other natural transformation 𝑏 sending to 𝑔, i.e., define the morphism part 𝑃 such that (𝑎(𝑈𝑃𝑉)𝑖) = 𝑏 such that (𝑏‘𝑋) = 𝑔 given hypotheses of fuco23 49970. Such construction should be provable as a functor.

Given any 𝑃, it is a morphism part of a functor compatible with the object part, i.e., the functor composition, i.e., the restriction of ∘_func, iff both of the following hold.

1. It has the same form as df-fuco 49946 up to fuco23 49970, but ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) might be mapped to a different morphism in category 𝐸. See fucofulem2 49940 for some insights.

2. fuco22nat 49975, fucoid 49977, and fucoco 49986 are satisfied.

(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⟩)
fucofunc.c	⊢ (𝜑 → 𝐶 ∈ Cat)
fucofunc.d	⊢ (𝜑 → 𝐷 ∈ Cat)
fucofunc.e	⊢ (𝜑 → 𝐸 ∈ Cat)

**Assertion**
Ref	Expression
fucofunc	⊢ (𝜑 → 𝑂(𝑇 Func 𝑄)𝑃)

Proof of Theorem fucofunc Dummy variables 𝑚 𝑛 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucoco2.t	. . 3 ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
2	1	xpcfucbas 49881	. 2 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = (Base‘𝑇)
3		fucoco2.q	. . 3 ⊢ 𝑄 = (𝐶 FuncCat 𝐸)
4	3	fucbas 18010	. 2 ⊢ (𝐶 Func 𝐸) = (Base‘𝑄)
5		eqid 2765	. 2 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
6		eqid 2765	. . 3 ⊢ (𝐶 Nat 𝐸) = (𝐶 Nat 𝐸)
7	3, 6	fuchom 18011	. 2 ⊢ (𝐶 Nat 𝐸) = (Hom ‘𝑄)
8		eqid 2765	. 2 ⊢ (Id‘𝑇) = (Id‘𝑇)
9		eqid 2765	. 2 ⊢ (Id‘𝑄) = (Id‘𝑄)
10		eqid 2765	. 2 ⊢ (comp‘𝑇) = (comp‘𝑇)
11		eqid 2765	. 2 ⊢ (comp‘𝑄) = (comp‘𝑄)
12		eqid 2765	. . . 4 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
13		fucofunc.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
14		fucofunc.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
15	12, 13, 14	fuccat 18020	. . 3 ⊢ (𝜑 → (𝐷 FuncCat 𝐸) ∈ Cat)
16		eqid 2765	. . . 4 ⊢ (𝐶 FuncCat 𝐷) = (𝐶 FuncCat 𝐷)
17		fucofunc.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
18	16, 17, 13	fuccat 18020	. . 3 ⊢ (𝜑 → (𝐶 FuncCat 𝐷) ∈ Cat)
19	1, 15, 18	xpccat 18236	. 2 ⊢ (𝜑 → 𝑇 ∈ Cat)
20	3, 17, 14	fuccat 18020	. 2 ⊢ (𝜑 → 𝑄 ∈ Cat)
21		fucoco2.o	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
22		eqidd 2766	. . 3 ⊢ (𝜑 → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
23	17, 13, 14, 21, 22	fucof1 49951	. 2 ⊢ (𝜑 → 𝑂:((𝐷 Func 𝐸) × (𝐶 Func 𝐷))⟶(𝐶 Func 𝐸))
24	17, 13, 14, 21, 22	fucofn2 49953	. 2 ⊢ (𝜑 → 𝑃 Fn (((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) × ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
25	21	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑦 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))) → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
26		eqidd 2766	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑦 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))) → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
27		simprl 782	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑦 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))) → 𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
28		simprr 784	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑦 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))) → 𝑦 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
29	25, 1, 5, 26, 27, 28	fucof21 49976	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑦 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))) → (𝑥𝑃𝑦):(𝑥(Hom ‘𝑇)𝑦)⟶((𝑂‘𝑥)(𝐶 Nat 𝐸)(𝑂‘𝑦)))
30	21	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
31		eqidd 2766	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
32		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) → 𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
33	30, 1, 8, 3, 9, 31, 32	fucoid2 49978	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) → ((𝑥𝑃𝑥)‘((Id‘𝑇)‘𝑥)) = ((Id‘𝑄)‘(𝑂‘𝑥)))
34	21	3ad2ant1 1149	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑦 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑧 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ∧ (𝑚 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑛 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
35		eqidd 2766	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑦 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑧 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ∧ (𝑚 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑛 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
36		simp21 1223	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑦 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑧 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ∧ (𝑚 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑛 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
37		simp22 1224	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑦 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑧 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ∧ (𝑚 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑛 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑦 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
38		simp23 1225	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑦 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑧 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ∧ (𝑚 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑛 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑧 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
39		simp3l 1218	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑦 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑧 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ∧ (𝑚 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑛 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑚 ∈ (𝑥(Hom ‘𝑇)𝑦))
40		simp3r 1219	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑦 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑧 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ∧ (𝑚 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑛 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑛 ∈ (𝑦(Hom ‘𝑇)𝑧))
41	1, 3, 34, 10, 11, 35, 36, 37, 38, 5, 39, 40	fucoco2 49987	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑦 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑧 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ∧ (𝑚 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑛 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩(comp‘𝑄)(𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚)))
42	2, 4, 5, 7, 8, 9, 10, 11, 19, 20, 23, 24, 29, 33, 41	isfuncd 17912	1 ⊢ (𝜑 → 𝑂(𝑇 Func 𝑄)𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ⟨cop 4591 class class class wbr 5105 × cxp 5650 ‘cfv 6525 (class class class)co 7400 Hom chom 17311 compcco 17312 Catccat 17710 Idccid 17711 Func cfunc 17901 Nat cnat 17991 FuncCat cfuc 17992 ×_c cxpc 18214 ∘_F cfuco 49945

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-er 8682 df-map 8814 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-fz 13527 df-struct 17197 df-slot 17232 df-ndx 17244 df-base 17260 df-hom 17324 df-cco 17325 df-cat 17714 df-cid 17715 df-func 17905 df-cofu 17907 df-nat 17993 df-fuc 17994 df-xpc 18218 df-fuco 49946

This theorem is referenced by: fucofunca 49989

W3C validator