fucofunc - Mathbox for Zhi Wang

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Theorem fucofunc 49348

Description: The functor composition bifunctor is a functor. See also fucofunca 49349.

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 49462), for example, (SetCat‘1_o) (the trivial category, setc1oterm 49480), 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 49330. 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 49306 up to fuco23 49330, but ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) might be mapped to a different morphism in category 𝐸. See fucofulem2 49300 for some insights.

2. fuco22nat 49335, fucoid 49337, and fucoco 49346 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 49241	. 2 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = (Base‘𝑇)
3		fucoco2.q	. . 3 ⊢ 𝑄 = (𝐶 FuncCat 𝐸)
4	3	fucbas 17870	. 2 ⊢ (𝐶 Func 𝐸) = (Base‘𝑄)
5		eqid 2729	. 2 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
6		eqid 2729	. . 3 ⊢ (𝐶 Nat 𝐸) = (𝐶 Nat 𝐸)
7	3, 6	fuchom 17871	. 2 ⊢ (𝐶 Nat 𝐸) = (Hom ‘𝑄)
8		eqid 2729	. 2 ⊢ (Id‘𝑇) = (Id‘𝑇)
9		eqid 2729	. 2 ⊢ (Id‘𝑄) = (Id‘𝑄)
10		eqid 2729	. 2 ⊢ (comp‘𝑇) = (comp‘𝑇)
11		eqid 2729	. 2 ⊢ (comp‘𝑄) = (comp‘𝑄)
12		eqid 2729	. . . 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 17880	. . 3 ⊢ (𝜑 → (𝐷 FuncCat 𝐸) ∈ Cat)
16		eqid 2729	. . . 4 ⊢ (𝐶 FuncCat 𝐷) = (𝐶 FuncCat 𝐷)
17		fucofunc.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
18	16, 17, 13	fuccat 17880	. . 3 ⊢ (𝜑 → (𝐶 FuncCat 𝐷) ∈ Cat)
19	1, 15, 18	xpccat 18096	. 2 ⊢ (𝜑 → 𝑇 ∈ Cat)
20	3, 17, 14	fuccat 17880	. 2 ⊢ (𝜑 → 𝑄 ∈ Cat)
21		fucoco2.o	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
22		eqidd 2730	. . 3 ⊢ (𝜑 → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
23	17, 13, 14, 21, 22	fucof1 49311	. 2 ⊢ (𝜑 → 𝑂:((𝐷 Func 𝐸) × (𝐶 Func 𝐷))⟶(𝐶 Func 𝐸))
24	17, 13, 14, 21, 22	fucofn2 49313	. 2 ⊢ (𝜑 → 𝑃 Fn (((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) × ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
25	21	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∧ 𝑦 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))) → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
26		eqidd 2730	. . 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 770	. . 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 772	. . 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 49336	. 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 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
31		eqidd 2730	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
32		simpr 484	. . 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 49338	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) → ((𝑥𝑃𝑥)‘((Id‘𝑇)‘𝑥)) = ((Id‘𝑄)‘(𝑂‘𝑥)))
34	21	3ad2ant1 1133	. . 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 2730	. . 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 1207	. . 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 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))) → 𝑦 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
38		simp23 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))) → 𝑧 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
39		simp3l 1202	. . 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 1203	. . 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 49347	. 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 17772	1 ⊢ (𝜑 → 𝑂(𝑇 Func 𝑄)𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ⟨cop 4583 class class class wbr 5092 × cxp 5617 ‘cfv 6482 (class class class)co 7349 Hom chom 17172 compcco 17173 Catccat 17570 Idccid 17571 Func cfunc 17761 Nat cnat 17851 FuncCat cfuc 17852 ×_c cxpc 18074 ∘_F cfuco 49305

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 49306

This theorem is referenced by: fucofunca 49349

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