fucofunc - Mathbox for Zhi Wang

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Theorem fucofunc 49856

Description: The functor composition bifunctor is a functor. See also fucofunca 49857.

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 49970), for example, (SetCat‘1_o) (the trivial category, setc1oterm 49988), 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 49838. 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 49814 up to fuco23 49838, but ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) might be mapped to a different morphism in category 𝐸. See fucofulem2 49808 for some insights.

2. fuco22nat 49843, fucoid 49845, and fucoco 49854 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 49749	. 2 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = (Base‘𝑇)
3		fucoco2.q	. . 3 ⊢ 𝑄 = (𝐶 FuncCat 𝐸)
4	3	fucbas 17928	. 2 ⊢ (𝐶 Func 𝐸) = (Base‘𝑄)
5		eqid 2740	. 2 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
6		eqid 2740	. . 3 ⊢ (𝐶 Nat 𝐸) = (𝐶 Nat 𝐸)
7	3, 6	fuchom 17929	. 2 ⊢ (𝐶 Nat 𝐸) = (Hom ‘𝑄)
8		eqid 2740	. 2 ⊢ (Id‘𝑇) = (Id‘𝑇)
9		eqid 2740	. 2 ⊢ (Id‘𝑄) = (Id‘𝑄)
10		eqid 2740	. 2 ⊢ (comp‘𝑇) = (comp‘𝑇)
11		eqid 2740	. 2 ⊢ (comp‘𝑄) = (comp‘𝑄)
12		eqid 2740	. . . 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 17938	. . 3 ⊢ (𝜑 → (𝐷 FuncCat 𝐸) ∈ Cat)
16		eqid 2740	. . . 4 ⊢ (𝐶 FuncCat 𝐷) = (𝐶 FuncCat 𝐷)
17		fucofunc.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
18	16, 17, 13	fuccat 17938	. . 3 ⊢ (𝜑 → (𝐶 FuncCat 𝐷) ∈ Cat)
19	1, 15, 18	xpccat 18154	. 2 ⊢ (𝜑 → 𝑇 ∈ Cat)
20	3, 17, 14	fuccat 17938	. 2 ⊢ (𝜑 → 𝑄 ∈ Cat)
21		fucoco2.o	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
22		eqidd 2741	. . 3 ⊢ (𝜑 → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
23	17, 13, 14, 21, 22	fucof1 49819	. 2 ⊢ (𝜑 → 𝑂:((𝐷 Func 𝐸) × (𝐶 Func 𝐷))⟶(𝐶 Func 𝐸))
24	17, 13, 14, 21, 22	fucofn2 49821	. 2 ⊢ (𝜑 → 𝑃 Fn (((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) × ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
25	21	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑦 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))) → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
26		eqidd 2741	. . 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 776	. . 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 778	. . 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 49844	. 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 481	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
31		eqidd 2741	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
32		simpr 485	. . 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 49846	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) → ((𝑥𝑃𝑥)‘((Id‘𝑇)‘𝑥)) = ((Id‘𝑄)‘(𝑂‘𝑥)))
34	21	3ad2ant1 1139	. . 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 2741	. . 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 1213	. . 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 1214	. . 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 1215	. . 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 1208	. . 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 1209	. . 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 49855	. 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 17830	1 ⊢ (𝜑 → 𝑂(𝑇 Func 𝑄)𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ⟨cop 4568 class class class wbr 5079 × cxp 5623 ‘cfv 6492 (class class class)co 7363 Hom chom 17229 compcco 17230 Catccat 17628 Idccid 17629 Func cfunc 17819 Nat cnat 17909 FuncCat cfuc 17910 ×_c cxpc 18132 ∘_F cfuco 49813

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-map 8772 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-fz 13460 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17178 df-hom 17242 df-cco 17243 df-cat 17632 df-cid 17633 df-func 17823 df-cofu 17825 df-nat 17911 df-fuc 17912 df-xpc 18136 df-fuco 49814

This theorem is referenced by: fucofunca 49857

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