Theorem diagffth 50040

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 49981	. . . 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 7983	. . 3 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝐿 ∈ (𝐶 Func 𝑄)) → 𝐿 = ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩)
9	1, 7, 8	sylancr 594	. 2 ⊢ (𝜑 → 𝐿 = ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩)
10	7	func1st2nd 49578	. . . 4 ⊢ (𝜑 → (1st ‘𝐿)(𝐶 Func 𝑄)(2nd ‘𝐿))
11		eqid 2741	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
12		eqid 2741	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
13		simprl 777	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
14		simprr 779	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
15		eqid 2741	. . . . . 6 ⊢ (𝐷 Nat 𝐶) = (𝐷 Nat 𝐶)
16	4	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐷 ∈ TermCat)
17	3	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
18	2, 11, 12, 13, 14, 15, 16, 17	diag2f1o 50039	. . . . 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 3184	. . . 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 590	. . 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 220	. 2 ⊢ (𝜑 → ⟨(1st ‘𝐿), (2nd ‘𝐿)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
25	9, 24	eqeltrd 2841	1 ⊢ (𝜑 → 𝐿 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055   ∩ cin 3883  ⟨cop 4563   class class class wbr 5074  Rel wrel 5625  –1-1-onto→wf1o 6487  ‘cfv 6488  (class class class)co 7359  1^st c1st 7931  2^nd c2nd 7932  Basecbs 17174  Hom chom 17226  Catccat 17625   Func cfunc 17816   Full cful 17866   Faith cfth 17867   Nat cnat 17906   FuncCat cfuc 17907  Δ_funccdiag 18173  TermCatctermc 49974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681  ax-cnex 11090  ax-resscn 11091  ax-1cn 11092  ax-icn 11093  ax-addcl 11094  ax-addrcl 11095  ax-mulcl 11096  ax-mulrcl 11097  ax-mulcom 11098  ax-addass 11099  ax-mulass 11100  ax-distr 11101  ax-i2m1 11102  ax-1ne0 11103  ax-1rid 11104  ax-rnegex 11105  ax-rrecex 11106  ax-cnre 11107  ax-pre-lttri 11108  ax-pre-lttrn 11109  ax-pre-ltadd 11110  ax-pre-mulgt0 11111

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  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 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7316  df-ov 7362  df-oprab 7363  df-mpo 7364  df-om 7810  df-1st 7933  df-2nd 7934  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-pnf 11177  df-mnf 11178  df-xr 11179  df-ltxr 11180  df-le 11181  df-sub 11375  df-neg 11376  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-dec 12640  df-uz 12784  df-fz 13457  df-struct 17112  df-slot 17147  df-ndx 17159  df-base 17175  df-hom 17239  df-cco 17240  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 49920  df-termc 49975

This theorem is referenced by:  diagciso  50041  lmdran  50173  cmdlan  50174