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Theorem ressffth 16820
Description: The inclusion functor from a full subcategory is a full and faithful functor, see also remark 4.4(2) in [Adamek] p. 49. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
ressffth.d 𝐷 = (𝐶s 𝑆)
ressffth.i 𝐼 = (idfunc𝐷)
Assertion
Ref Expression
ressffth ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))

Proof of Theorem ressffth
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfunc 16744 . . 3 Rel (𝐷 Func 𝐷)
2 ressffth.d . . . . 5 𝐷 = (𝐶s 𝑆)
3 resscat 16734 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐶s 𝑆) ∈ Cat)
42, 3syl5eqel 2844 . . . 4 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐷 ∈ Cat)
5 ressffth.i . . . . 5 𝐼 = (idfunc𝐷)
65idfucl 16763 . . . 4 (𝐷 ∈ Cat → 𝐼 ∈ (𝐷 Func 𝐷))
74, 6syl 17 . . 3 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 ∈ (𝐷 Func 𝐷))
8 1st2nd 7383 . . 3 ((Rel (𝐷 Func 𝐷) ∧ 𝐼 ∈ (𝐷 Func 𝐷)) → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
91, 7, 8sylancr 698 . 2 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
10 eqidd 2762 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf𝐷) = (Homf𝐷))
11 eqidd 2762 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf𝐷) = (compf𝐷))
12 eqid 2761 . . . . . . . . . . . . . 14 (Base‘𝐶) = (Base‘𝐶)
1312ressinbas 16159 . . . . . . . . . . . . 13 (𝑆𝑉 → (𝐶s 𝑆) = (𝐶s (𝑆 ∩ (Base‘𝐶))))
1413adantl 473 . . . . . . . . . . . 12 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐶s 𝑆) = (𝐶s (𝑆 ∩ (Base‘𝐶))))
152, 14syl5eq 2807 . . . . . . . . . . 11 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐷 = (𝐶s (𝑆 ∩ (Base‘𝐶))))
1615fveq2d 6358 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf𝐷) = (Homf ‘(𝐶s (𝑆 ∩ (Base‘𝐶)))))
17 eqid 2761 . . . . . . . . . . . 12 (Homf𝐶) = (Homf𝐶)
18 simpl 474 . . . . . . . . . . . 12 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐶 ∈ Cat)
19 inss2 3978 . . . . . . . . . . . . 13 (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶)
2019a1i 11 . . . . . . . . . . . 12 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶))
21 eqid 2761 . . . . . . . . . . . 12 (𝐶s (𝑆 ∩ (Base‘𝐶))) = (𝐶s (𝑆 ∩ (Base‘𝐶)))
22 eqid 2761 . . . . . . . . . . . 12 (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶))))) = (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))
2312, 17, 18, 20, 21, 22fullresc 16733 . . . . . . . . . . 11 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → ((Homf ‘(𝐶s (𝑆 ∩ (Base‘𝐶)))) = (Homf ‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))) ∧ (compf‘(𝐶s (𝑆 ∩ (Base‘𝐶)))) = (compf‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶))))))))
2423simpld 477 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf ‘(𝐶s (𝑆 ∩ (Base‘𝐶)))) = (Homf ‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
2516, 24eqtrd 2795 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf𝐷) = (Homf ‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
2615fveq2d 6358 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf𝐷) = (compf‘(𝐶s (𝑆 ∩ (Base‘𝐶)))))
2723simprd 482 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf‘(𝐶s (𝑆 ∩ (Base‘𝐶)))) = (compf‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
2826, 27eqtrd 2795 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf𝐷) = (compf‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
29 ovex 6843 . . . . . . . . . . 11 (𝐶s 𝑆) ∈ V
302, 29eqeltri 2836 . . . . . . . . . 10 𝐷 ∈ V
3130a1i 11 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐷 ∈ V)
32 ovex 6843 . . . . . . . . . 10 (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶))))) ∈ V
3332a1i 11 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶))))) ∈ V)
3410, 11, 25, 28, 31, 31, 31, 33funcpropd 16782 . . . . . . . 8 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐷 Func 𝐷) = (𝐷 Func (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
3512, 17, 18, 20fullsubc 16732 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))) ∈ (Subcat‘𝐶))
36 funcres2 16780 . . . . . . . . 9 (((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))) ∈ (Subcat‘𝐶) → (𝐷 Func (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))) ⊆ (𝐷 Func 𝐶))
3735, 36syl 17 . . . . . . . 8 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐷 Func (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))) ⊆ (𝐷 Func 𝐶))
3834, 37eqsstrd 3781 . . . . . . 7 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐷 Func 𝐷) ⊆ (𝐷 Func 𝐶))
3938, 7sseldd 3746 . . . . . 6 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 ∈ (𝐷 Func 𝐶))
409, 39eqeltrrd 2841 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → ⟨(1st𝐼), (2nd𝐼)⟩ ∈ (𝐷 Func 𝐶))
41 df-br 4806 . . . . 5 ((1st𝐼)(𝐷 Func 𝐶)(2nd𝐼) ↔ ⟨(1st𝐼), (2nd𝐼)⟩ ∈ (𝐷 Func 𝐶))
4240, 41sylibr 224 . . . 4 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (1st𝐼)(𝐷 Func 𝐶)(2nd𝐼))
43 f1oi 6337 . . . . . 6 ( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(𝑥(Hom ‘𝐷)𝑦)
44 eqid 2761 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
454adantr 472 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐷 ∈ Cat)
46 eqid 2761 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
47 simprl 811 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
48 simprr 813 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
495, 44, 45, 46, 47, 48idfu2nd 16759 . . . . . . 7 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(2nd𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
50 eqidd 2762 . . . . . . 7 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
51 eqid 2761 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
522, 51resshom 16301 . . . . . . . . 9 (𝑆𝑉 → (Hom ‘𝐶) = (Hom ‘𝐷))
5352ad2antlr 765 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Hom ‘𝐶) = (Hom ‘𝐷))
545, 44, 45, 47idfu1 16762 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st𝐼)‘𝑥) = 𝑥)
555, 44, 45, 48idfu1 16762 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st𝐼)‘𝑦) = 𝑦)
5653, 54, 55oveq123d 6836 . . . . . . 7 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (((1st𝐼)‘𝑥)(Hom ‘𝐶)((1st𝐼)‘𝑦)) = (𝑥(Hom ‘𝐷)𝑦))
5749, 50, 56f1oeq123d 6296 . . . . . 6 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((𝑥(2nd𝐼)𝑦):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(((1st𝐼)‘𝑥)(Hom ‘𝐶)((1st𝐼)‘𝑦)) ↔ ( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(𝑥(Hom ‘𝐷)𝑦)))
5843, 57mpbiri 248 . . . . 5 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(2nd𝐼)𝑦):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(((1st𝐼)‘𝑥)(Hom ‘𝐶)((1st𝐼)‘𝑦)))
5958ralrimivva 3110 . . . 4 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → ∀𝑥 ∈ (Base‘𝐷)∀𝑦 ∈ (Base‘𝐷)(𝑥(2nd𝐼)𝑦):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(((1st𝐼)‘𝑥)(Hom ‘𝐶)((1st𝐼)‘𝑦)))
6044, 46, 51isffth2 16798 . . . 4 ((1st𝐼)((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶))(2nd𝐼) ↔ ((1st𝐼)(𝐷 Func 𝐶)(2nd𝐼) ∧ ∀𝑥 ∈ (Base‘𝐷)∀𝑦 ∈ (Base‘𝐷)(𝑥(2nd𝐼)𝑦):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(((1st𝐼)‘𝑥)(Hom ‘𝐶)((1st𝐼)‘𝑦))))
6142, 59, 60sylanbrc 701 . . 3 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (1st𝐼)((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶))(2nd𝐼))
62 df-br 4806 . . 3 ((1st𝐼)((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶))(2nd𝐼) ↔ ⟨(1st𝐼), (2nd𝐼)⟩ ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
6361, 62sylib 208 . 2 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → ⟨(1st𝐼), (2nd𝐼)⟩ ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
649, 63eqeltrd 2840 1 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2140  wral 3051  Vcvv 3341  cin 3715  wss 3716  cop 4328   class class class wbr 4805   I cid 5174   × cxp 5265  cres 5269  Rel wrel 5272  1-1-ontowf1o 6049  cfv 6050  (class class class)co 6815  1st c1st 7333  2nd c2nd 7334  Basecbs 16080  s cress 16081  Hom chom 16175  Catccat 16547  Homf chomf 16549  compfccomf 16550  cat cresc 16690  Subcatcsubc 16691   Func cfunc 16736  idfunccidfu 16737   Full cful 16784   Faith cfth 16785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-mulcom 10213  ax-addass 10214  ax-mulass 10215  ax-distr 10216  ax-i2m1 10217  ax-1ne0 10218  ax-1rid 10219  ax-rnegex 10220  ax-rrecex 10221  ax-cnre 10222  ax-pre-lttri 10223  ax-pre-lttrn 10224  ax-pre-ltadd 10225  ax-pre-mulgt0 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-1st 7335  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-er 7914  df-map 8028  df-pm 8029  df-ixp 8078  df-en 8125  df-dom 8126  df-sdom 8127  df-pnf 10289  df-mnf 10290  df-xr 10291  df-ltxr 10292  df-le 10293  df-sub 10481  df-neg 10482  df-nn 11234  df-2 11292  df-3 11293  df-4 11294  df-5 11295  df-6 11296  df-7 11297  df-8 11298  df-9 11299  df-n0 11506  df-z 11591  df-dec 11707  df-ndx 16083  df-slot 16084  df-base 16086  df-sets 16087  df-ress 16088  df-hom 16189  df-cco 16190  df-cat 16551  df-cid 16552  df-homf 16553  df-comf 16554  df-ssc 16692  df-resc 16693  df-subc 16694  df-func 16740  df-idfu 16741  df-full 16786  df-fth 16787
This theorem is referenced by: (None)
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