Theorem diagffth 50344

Description: The diagonal functor is a fully faithful functor from a category 𝐶 to the category of functors from a terminal category to 𝐶. (Contributed by Zhi Wang, 21-Oct-2025.)

Hypotheses

Ref	Expression
diagffth.c	⊢ (𝜑 → 𝐶 ∈ Cat)
diagffth.d	⊢ (𝜑 → 𝐷 ∈ TermCat)
diagffth.q	⊢ 𝑄 = (𝐷 FuncCat 𝐶)
diagffth.l	⊢ 𝐿 = (𝐶Δfunc𝐷)

Assertion

Ref	Expression
diagffth	⊢ (𝜑 → 𝐿 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))

Proof of Theorem diagffth

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relfunc 17925	. . 3 ⊢ Rel (𝐶 Func 𝑄)
2		diagffth.l	. . . 4 ⊢ 𝐿 = (𝐶Δfunc𝐷)
3		diagffth.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		diagffth.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ TermCat)
5	4	termccd 50285	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
6		diagffth.q	. . . 4 ⊢ 𝑄 = (𝐷 FuncCat 𝐶)
7	2, 3, 5, 6	diagcl 18303	. . 3 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func 𝑄))
8		1st2nd 8034	. . 3 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝐿 ∈ (𝐶 Func 𝑄)) → 𝐿 = ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩)
9	1, 7, 8	sylancr 598	. 2 ⊢ (𝜑 → 𝐿 = ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩)
10	7	func1st2nd 49882	. . . 4 ⊢ (𝜑 → (1st ‘𝐿)(𝐶 Func 𝑄)(2nd ‘𝐿))
11		eqid 2762	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
12		eqid 2762	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
13		simprl 782	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
14		simprr 784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
15		eqid 2762	. . . . . 6 ⊢ (𝐷 Nat 𝐶) = (𝐷 Nat 𝐶)
16	4	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐷 ∈ TermCat)
17	3	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
18	2, 11, 12, 13, 14, 15, 16, 17	diag2f1o 50343	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝐿)𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(((1st ‘𝐿)‘𝑥)(𝐷 Nat 𝐶)((1st ‘𝐿)‘𝑦)))
19	18	ralrimivva 3207	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(2nd ‘𝐿)𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(((1st ‘𝐿)‘𝑥)(𝐷 Nat 𝐶)((1st ‘𝐿)‘𝑦)))
20	6, 15	fuchom 18027	. . . . 5 ⊢ (𝐷 Nat 𝐶) = (Hom ‘𝑄)
21	11, 12, 20	isffth2 17981	. . . 4 ⊢ ((1st ‘𝐿)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝐿) ↔ ((1st ‘𝐿)(𝐶 Func 𝑄)(2nd ‘𝐿) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(2nd ‘𝐿)𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1-onto→(((1st ‘𝐿)‘𝑥)(𝐷 Nat 𝐶)((1st ‘𝐿)‘𝑦))))
22	10, 19, 21	sylanbrc 594	. . 3 ⊢ (𝜑 → (1st ‘𝐿)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝐿))
23		df-br 5109	. . 3 ⊢ ((1st ‘𝐿)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝐿) ↔ ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
24	22, 23	sylib 221	. 2 ⊢ (𝜑 → ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
25	9, 24	eqeltrd 2862	1 ⊢ (𝜑 → 𝐿 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078   ∩ cin 3903  ⟨cop 4594   class class class wbr 5108  Rel wrel 5665  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7412  1^st c1st 7982  2^nd c2nd 7983  Basecbs 17275  Hom chom 17327  Catccat 17726   Func cfunc 17917   Full cful 17967   Faith cfth 17968   Nat cnat 18007   FuncCat cfuc 18008  Δ_funccdiag 18274  TermCatctermc 50278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-er 8692  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-2 12309  df-3 12310  df-4 12311  df-5 12312  df-6 12313  df-7 12314  df-8 12315  df-9 12316  df-n0 12511  df-z 12598  df-dec 12718  df-uz 12869  df-fz 13542  df-struct 17213  df-slot 17248  df-ndx 17260  df-base 17276  df-hom 17340  df-cco 17341  df-cat 17730  df-cid 17731  df-func 17921  df-full 17969  df-fth 17970  df-nat 18009  df-fuc 18010  df-xpc 18234  df-1stf 18235  df-curf 18276  df-diag 18278  df-thinc 50224  df-termc 50279

This theorem is used by:  diagciso  50345  lmdran  50477  cmdlan  50478