Theorem diagffth 50001

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 17818	. . 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 49942	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
6		diagffth.q	. . . 4 ⊢ 𝑄 = (𝐷 FuncCat 𝐶)
7	2, 3, 5, 6	diagcl 18196	. . 3 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func 𝑄))
8		1st2nd 7981	. . 3 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝐿 ∈ (𝐶 Func 𝑄)) → 𝐿 = ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩)
9	1, 7, 8	sylancr 588	. 2 ⊢ (𝜑 → 𝐿 = ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩)
10	7	func1st2nd 49539	. . . 4 ⊢ (𝜑 → (1st ‘𝐿)(𝐶 Func 𝑄)(2nd ‘𝐿))
11		eqid 2735	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
12		eqid 2735	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
13		simprl 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
14		simprr 773	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
15		eqid 2735	. . . . . 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 50000	. . . . 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 3178	. . . 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 17920	. . . . 5 ⊢ (𝐷 Nat 𝐶) = (Hom ‘𝑄)
21	11, 12, 20	isffth2 17874	. . . 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 584	. . 3 ⊢ (𝜑 → (1st ‘𝐿)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝐿))
23		df-br 5075	. . 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 2835	1 ⊢ (𝜑 → 𝐿 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3049   ∩ cin 3884  ⟨cop 4563   class class class wbr 5074  Rel wrel 5625  –1-1-onto→wf1o 6486  ‘cfv 6487  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  Basecbs 17168  Hom chom 17220  Catccat 17619   Func cfunc 17810   Full cful 17860   Faith cfth 17861   Nat cnat 17900   FuncCat cfuc 17901  Δ_funccdiag 18167  TermCatctermc 49935

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 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8632  df-map 8764  df-ixp 8835  df-en 8883  df-dom 8884  df-sdom 8885  df-fin 8886  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12164  df-2 12233  df-3 12234  df-4 12235  df-5 12236  df-6 12237  df-7 12238  df-8 12239  df-9 12240  df-n0 12427  df-z 12514  df-dec 12634  df-uz 12778  df-fz 13451  df-struct 17106  df-slot 17141  df-ndx 17153  df-base 17169  df-hom 17233  df-cco 17234  df-cat 17623  df-cid 17624  df-func 17814  df-full 17862  df-fth 17863  df-nat 17902  df-fuc 17903  df-xpc 18127  df-1stf 18128  df-curf 18169  df-diag 18171  df-thinc 49881  df-termc 49936

This theorem is referenced by:  diagciso  50002  lmdran  50134  cmdlan  50135