fucofunc - Mathbox for Zhi Wang

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Theorem fucofunc 49712

Description: The functor composition bifunctor is a functor. See also fucofunca 49713.

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 49826), for example, (SetCat‘1_o) (the trivial category, setc1oterm 49844), 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 49694. 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 49670 up to fuco23 49694, but ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) might be mapped to a different morphism in category 𝐸. See fucofulem2 49664 for some insights.

2. fuco22nat 49699, fucoid 49701, and fucoco 49710 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 49605	. 2 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = (Base‘𝑇)
3		fucoco2.q	. . 3 ⊢ 𝑄 = (𝐶 FuncCat 𝐸)
4	3	fucbas 17899	. 2 ⊢ (𝐶 Func 𝐸) = (Base‘𝑄)
5		eqid 2737	. 2 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
6		eqid 2737	. . 3 ⊢ (𝐶 Nat 𝐸) = (𝐶 Nat 𝐸)
7	3, 6	fuchom 17900	. 2 ⊢ (𝐶 Nat 𝐸) = (Hom ‘𝑄)
8		eqid 2737	. 2 ⊢ (Id‘𝑇) = (Id‘𝑇)
9		eqid 2737	. 2 ⊢ (Id‘𝑄) = (Id‘𝑄)
10		eqid 2737	. 2 ⊢ (comp‘𝑇) = (comp‘𝑇)
11		eqid 2737	. 2 ⊢ (comp‘𝑄) = (comp‘𝑄)
12		eqid 2737	. . . 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 17909	. . 3 ⊢ (𝜑 → (𝐷 FuncCat 𝐸) ∈ Cat)
16		eqid 2737	. . . 4 ⊢ (𝐶 FuncCat 𝐷) = (𝐶 FuncCat 𝐷)
17		fucofunc.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
18	16, 17, 13	fuccat 17909	. . 3 ⊢ (𝜑 → (𝐶 FuncCat 𝐷) ∈ Cat)
19	1, 15, 18	xpccat 18125	. 2 ⊢ (𝜑 → 𝑇 ∈ Cat)
20	3, 17, 14	fuccat 17909	. 2 ⊢ (𝜑 → 𝑄 ∈ Cat)
21		fucoco2.o	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
22		eqidd 2738	. . 3 ⊢ (𝜑 → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
23	17, 13, 14, 21, 22	fucof1 49675	. 2 ⊢ (𝜑 → 𝑂:((𝐷 Func 𝐸) × (𝐶 Func 𝐷))⟶(𝐶 Func 𝐸))
24	17, 13, 14, 21, 22	fucofn2 49677	. 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 2738	. . 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 771	. . 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 773	. . 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 49700	. 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 2738	. . 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 49702	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) → ((𝑥𝑃𝑥)‘((Id‘𝑇)‘𝑥)) = ((Id‘𝑄)‘(𝑂‘𝑥)))
34	21	3ad2ant1 1134	. . 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 2738	. . 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 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))) → 𝑥 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
37		simp22 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))) → 𝑦 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
38		simp23 1210	. . 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 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))) → 𝑚 ∈ (𝑥(Hom ‘𝑇)𝑦))
40		simp3r 1204	. . 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 49711	. 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 17801	1 ⊢ (𝜑 → 𝑂(𝑇 Func 𝑄)𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ⟨cop 4588 class class class wbr 5100 × cxp 5630 ‘cfv 6500 (class class class)co 7368 Hom chom 17200 compcco 17201 Catccat 17599 Idccid 17600 Func cfunc 17790 Nat cnat 17880 FuncCat cfuc 17881 ×_c cxpc 18103 ∘_F cfuco 49669

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 49670

This theorem is referenced by: fucofunca 49713

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