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Theorem ressffth 16805
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 16729 . . 3 Rel (𝐷 Func 𝐷)
2 ressffth.d . . . . 5 𝐷 = (𝐶s 𝑆)
3 resscat 16719 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐶s 𝑆) ∈ Cat)
42, 3syl5eqel 2854 . . . 4 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐷 ∈ Cat)
5 ressffth.i . . . . 5 𝐼 = (idfunc𝐷)
65idfucl 16748 . . . 4 (𝐷 ∈ Cat → 𝐼 ∈ (𝐷 Func 𝐷))
74, 6syl 17 . . 3 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 ∈ (𝐷 Func 𝐷))
8 1st2nd 7367 . . 3 ((Rel (𝐷 Func 𝐷) ∧ 𝐼 ∈ (𝐷 Func 𝐷)) → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
91, 7, 8sylancr 575 . 2 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
10 eqidd 2772 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf𝐷) = (Homf𝐷))
11 eqidd 2772 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf𝐷) = (compf𝐷))
12 eqid 2771 . . . . . . . . . . . . . 14 (Base‘𝐶) = (Base‘𝐶)
1312ressinbas 16143 . . . . . . . . . . . . 13 (𝑆𝑉 → (𝐶s 𝑆) = (𝐶s (𝑆 ∩ (Base‘𝐶))))
1413adantl 467 . . . . . . . . . . . 12 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐶s 𝑆) = (𝐶s (𝑆 ∩ (Base‘𝐶))))
152, 14syl5eq 2817 . . . . . . . . . . 11 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐷 = (𝐶s (𝑆 ∩ (Base‘𝐶))))
1615fveq2d 6337 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf𝐷) = (Homf ‘(𝐶s (𝑆 ∩ (Base‘𝐶)))))
17 eqid 2771 . . . . . . . . . . . 12 (Homf𝐶) = (Homf𝐶)
18 simpl 468 . . . . . . . . . . . 12 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐶 ∈ Cat)
19 inss2 3982 . . . . . . . . . . . . 13 (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶)
2019a1i 11 . . . . . . . . . . . 12 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶))
21 eqid 2771 . . . . . . . . . . . 12 (𝐶s (𝑆 ∩ (Base‘𝐶))) = (𝐶s (𝑆 ∩ (Base‘𝐶)))
22 eqid 2771 . . . . . . . . . . . 12 (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶))))) = (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))
2312, 17, 18, 20, 21, 22fullresc 16718 . . . . . . . . . . 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 482 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf ‘(𝐶s (𝑆 ∩ (Base‘𝐶)))) = (Homf ‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
2516, 24eqtrd 2805 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf𝐷) = (Homf ‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
2615fveq2d 6337 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf𝐷) = (compf‘(𝐶s (𝑆 ∩ (Base‘𝐶)))))
2723simprd 483 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf‘(𝐶s (𝑆 ∩ (Base‘𝐶)))) = (compf‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
2826, 27eqtrd 2805 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf𝐷) = (compf‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
29 ovex 6827 . . . . . . . . . . 11 (𝐶s 𝑆) ∈ V
302, 29eqeltri 2846 . . . . . . . . . 10 𝐷 ∈ V
3130a1i 11 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐷 ∈ V)
32 ovex 6827 . . . . . . . . . 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 16767 . . . . . . . 8 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐷 Func 𝐷) = (𝐷 Func (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
3512, 17, 18, 20fullsubc 16717 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))) ∈ (Subcat‘𝐶))
36 funcres2 16765 . . . . . . . . 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 3788 . . . . . . 7 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐷 Func 𝐷) ⊆ (𝐷 Func 𝐶))
3938, 7sseldd 3753 . . . . . 6 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 ∈ (𝐷 Func 𝐶))
409, 39eqeltrrd 2851 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → ⟨(1st𝐼), (2nd𝐼)⟩ ∈ (𝐷 Func 𝐶))
41 df-br 4788 . . . . 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 6316 . . . . . 6 ( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(𝑥(Hom ‘𝐷)𝑦)
44 eqid 2771 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
454adantr 466 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐷 ∈ Cat)
46 eqid 2771 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
47 simprl 754 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
48 simprr 756 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
495, 44, 45, 46, 47, 48idfu2nd 16744 . . . . . . 7 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(2nd𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
50 eqidd 2772 . . . . . . 7 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
51 eqid 2771 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
522, 51resshom 16286 . . . . . . . . 9 (𝑆𝑉 → (Hom ‘𝐶) = (Hom ‘𝐷))
5352ad2antlr 706 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Hom ‘𝐶) = (Hom ‘𝐷))
545, 44, 45, 47idfu1 16747 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st𝐼)‘𝑥) = 𝑥)
555, 44, 45, 48idfu1 16747 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st𝐼)‘𝑦) = 𝑦)
5653, 54, 55oveq123d 6817 . . . . . . 7 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (((1st𝐼)‘𝑥)(Hom ‘𝐶)((1st𝐼)‘𝑦)) = (𝑥(Hom ‘𝐷)𝑦))
5749, 50, 56f1oeq123d 6275 . . . . . 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 3120 . . . 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 16783 . . . 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 572 . . 3 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (1st𝐼)((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶))(2nd𝐼))
62 df-br 4788 . . 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 2850 1 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  cin 3722  wss 3723  cop 4323   class class class wbr 4787   I cid 5157   × cxp 5248  cres 5252  Rel wrel 5255  1-1-ontowf1o 6029  cfv 6030  (class class class)co 6796  1st c1st 7317  2nd c2nd 7318  Basecbs 16064  s cress 16065  Hom chom 16160  Catccat 16532  Homf chomf 16534  compfccomf 16535  cat cresc 16675  Subcatcsubc 16676   Func cfunc 16721  idfunccidfu 16722   Full cful 16769   Faith cfth 16770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-er 7900  df-map 8015  df-pm 8016  df-ixp 8067  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  df-3 11286  df-4 11287  df-5 11288  df-6 11289  df-7 11290  df-8 11291  df-9 11292  df-n0 11500  df-z 11585  df-dec 11701  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-hom 16174  df-cco 16175  df-cat 16536  df-cid 16537  df-homf 16538  df-comf 16539  df-ssc 16677  df-resc 16678  df-subc 16679  df-func 16725  df-idfu 16726  df-full 16771  df-fth 16772
This theorem is referenced by: (None)
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