Theorem diagffth 49543

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 17769	. . 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 49484	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
6		diagffth.q	. . . 4 ⊢ 𝑄 = (𝐷 FuncCat 𝐶)
7	2, 3, 5, 6	diagcl 18147	. . 3 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func 𝑄))
8		1st2nd 7974	. . 3 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝐿 ∈ (𝐶 Func 𝑄)) → 𝐿 = ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩)
9	1, 7, 8	sylancr 587	. 2 ⊢ (𝜑 → 𝐿 = ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩)
10	7	func1st2nd 49081	. . . 4 ⊢ (𝜑 → (1st ‘𝐿)(𝐶 Func 𝑄)(2nd ‘𝐿))
11		eqid 2729	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
12		eqid 2729	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
13		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
14		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
15		eqid 2729	. . . . . 6 ⊢ (𝐷 Nat 𝐶) = (𝐷 Nat 𝐶)
16	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐷 ∈ TermCat)
17	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
18	2, 11, 12, 13, 14, 15, 16, 17	diag2f1o 49542	. . . . 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 3172	. . . 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 17871	. . . . 5 ⊢ (𝐷 Nat 𝐶) = (Hom ‘𝑄)
21	11, 12, 20	isffth2 17825	. . . 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 583	. . 3 ⊢ (𝜑 → (1st ‘𝐿)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝐿))
23		df-br 5093	. . 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 218	. 2 ⊢ (𝜑 → ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
25	9, 24	eqeltrd 2828	1 ⊢ (𝜑 → 𝐿 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∩ cin 3902  ⟨cop 4583   class class class wbr 5092  Rel wrel 5624  –1-1-onto→wf1o 6481  ‘cfv 6482  (class class class)co 7349  1^st c1st 7922  2^nd c2nd 7923  Basecbs 17120  Hom chom 17172  Catccat 17570   Func cfunc 17761   Full cful 17811   Faith cfth 17812   Nat cnat 17851   FuncCat cfuc 17852  Δ_funccdiag 18118  TermCatctermc 49477

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-full 17813  df-fth 17814  df-nat 17853  df-fuc 17854  df-xpc 18078  df-1stf 18079  df-curf 18120  df-diag 18122  df-thinc 49423  df-termc 49478

This theorem is referenced by:  diagciso  49544  lmdran  49676  cmdlan  49677