Theorem diagffth 49524

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 17824	. . 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 49465	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
6		diagffth.q	. . . 4 ⊢ 𝑄 = (𝐷 FuncCat 𝐶)
7	2, 3, 5, 6	diagcl 18202	. . 3 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func 𝑄))
8		1st2nd 8018	. . 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 49062	. . . 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 49523	. . . . 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 3180	. . . 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 17926	. . . . 5 ⊢ (𝐷 Nat 𝐶) = (Hom ‘𝑄)
21	11, 12, 20	isffth2 17880	. . . 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 5108	. . 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 3913  ⟨cop 4595   class class class wbr 5107  Rel wrel 5643  –1-1-onto→wf1o 6510  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967  Basecbs 17179  Hom chom 17231  Catccat 17625   Func cfunc 17816   Full cful 17866   Faith cfth 17867   Nat cnat 17906   FuncCat cfuc 17907  Δ_funccdiag 18173  TermCatctermc 49458

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-hom 17244  df-cco 17245  df-cat 17629  df-cid 17630  df-func 17820  df-full 17868  df-fth 17869  df-nat 17908  df-fuc 17909  df-xpc 18133  df-1stf 18134  df-curf 18175  df-diag 18177  df-thinc 49404  df-termc 49459

This theorem is referenced by:  diagciso  49525  lmdran  49657  cmdlan  49658