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Theorem ressffth 16363
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 16287 . . 3 Rel (𝐷 Func 𝐷)
2 ressffth.d . . . . 5 𝐷 = (𝐶s 𝑆)
3 resscat 16277 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐶s 𝑆) ∈ Cat)
42, 3syl5eqel 2687 . . . 4 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐷 ∈ Cat)
5 ressffth.i . . . . 5 𝐼 = (idfunc𝐷)
65idfucl 16306 . . . 4 (𝐷 ∈ Cat → 𝐼 ∈ (𝐷 Func 𝐷))
74, 6syl 17 . . 3 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 ∈ (𝐷 Func 𝐷))
8 1st2nd 7078 . . 3 ((Rel (𝐷 Func 𝐷) ∧ 𝐼 ∈ (𝐷 Func 𝐷)) → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
91, 7, 8sylancr 693 . 2 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
10 eqidd 2606 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf𝐷) = (Homf𝐷))
11 eqidd 2606 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf𝐷) = (compf𝐷))
12 eqid 2605 . . . . . . . . . . . . . 14 (Base‘𝐶) = (Base‘𝐶)
1312ressinbas 15705 . . . . . . . . . . . . 13 (𝑆𝑉 → (𝐶s 𝑆) = (𝐶s (𝑆 ∩ (Base‘𝐶))))
1413adantl 480 . . . . . . . . . . . 12 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐶s 𝑆) = (𝐶s (𝑆 ∩ (Base‘𝐶))))
152, 14syl5eq 2651 . . . . . . . . . . 11 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐷 = (𝐶s (𝑆 ∩ (Base‘𝐶))))
1615fveq2d 6088 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf𝐷) = (Homf ‘(𝐶s (𝑆 ∩ (Base‘𝐶)))))
17 eqid 2605 . . . . . . . . . . . 12 (Homf𝐶) = (Homf𝐶)
18 simpl 471 . . . . . . . . . . . 12 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐶 ∈ Cat)
19 inss2 3791 . . . . . . . . . . . . 13 (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶)
2019a1i 11 . . . . . . . . . . . 12 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶))
21 eqid 2605 . . . . . . . . . . . 12 (𝐶s (𝑆 ∩ (Base‘𝐶))) = (𝐶s (𝑆 ∩ (Base‘𝐶)))
22 eqid 2605 . . . . . . . . . . . 12 (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶))))) = (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))
2312, 17, 18, 20, 21, 22fullresc 16276 . . . . . . . . . . 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 473 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf ‘(𝐶s (𝑆 ∩ (Base‘𝐶)))) = (Homf ‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
2516, 24eqtrd 2639 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf𝐷) = (Homf ‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
2615fveq2d 6088 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf𝐷) = (compf‘(𝐶s (𝑆 ∩ (Base‘𝐶)))))
2723simprd 477 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf‘(𝐶s (𝑆 ∩ (Base‘𝐶)))) = (compf‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
2826, 27eqtrd 2639 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf𝐷) = (compf‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
29 ovex 6551 . . . . . . . . . . 11 (𝐶s 𝑆) ∈ V
302, 29eqeltri 2679 . . . . . . . . . 10 𝐷 ∈ V
3130a1i 11 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐷 ∈ V)
32 ovex 6551 . . . . . . . . . 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 16325 . . . . . . . 8 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐷 Func 𝐷) = (𝐷 Func (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
3512, 17, 18, 20fullsubc 16275 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))) ∈ (Subcat‘𝐶))
36 funcres2 16323 . . . . . . . . 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 3597 . . . . . . 7 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐷 Func 𝐷) ⊆ (𝐷 Func 𝐶))
3938, 7sseldd 3564 . . . . . 6 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 ∈ (𝐷 Func 𝐶))
409, 39eqeltrrd 2684 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → ⟨(1st𝐼), (2nd𝐼)⟩ ∈ (𝐷 Func 𝐶))
41 df-br 4574 . . . . 5 ((1st𝐼)(𝐷 Func 𝐶)(2nd𝐼) ↔ ⟨(1st𝐼), (2nd𝐼)⟩ ∈ (𝐷 Func 𝐶))
4240, 41sylibr 222 . . . 4 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (1st𝐼)(𝐷 Func 𝐶)(2nd𝐼))
43 f1oi 6067 . . . . . 6 ( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(𝑥(Hom ‘𝐷)𝑦)
44 eqid 2605 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
454adantr 479 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐷 ∈ Cat)
46 eqid 2605 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
47 simprl 789 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
48 simprr 791 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
495, 44, 45, 46, 47, 48idfu2nd 16302 . . . . . . 7 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(2nd𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
50 eqidd 2606 . . . . . . 7 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
51 eqid 2605 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
522, 51resshom 15843 . . . . . . . . 9 (𝑆𝑉 → (Hom ‘𝐶) = (Hom ‘𝐷))
5352ad2antlr 758 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Hom ‘𝐶) = (Hom ‘𝐷))
545, 44, 45, 47idfu1 16305 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st𝐼)‘𝑥) = 𝑥)
555, 44, 45, 48idfu1 16305 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st𝐼)‘𝑦) = 𝑦)
5653, 54, 55oveq123d 6544 . . . . . . 7 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (((1st𝐼)‘𝑥)(Hom ‘𝐶)((1st𝐼)‘𝑦)) = (𝑥(Hom ‘𝐷)𝑦))
5749, 50, 56f1oeq123d 6027 . . . . . 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 246 . . . . 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 2949 . . . 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 16341 . . . 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 694 . . 3 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (1st𝐼)((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶))(2nd𝐼))
62 df-br 4574 . . 3 ((1st𝐼)((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶))(2nd𝐼) ↔ ⟨(1st𝐼), (2nd𝐼)⟩ ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
6361, 62sylib 206 . 2 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → ⟨(1st𝐼), (2nd𝐼)⟩ ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
649, 63eqeltrd 2683 1 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  wral 2891  Vcvv 3168  cin 3534  wss 3535  cop 4126   class class class wbr 4573   I cid 4934   × cxp 5022  cres 5026  Rel wrel 5029  1-1-ontowf1o 5785  cfv 5786  (class class class)co 6523  1st c1st 7030  2nd c2nd 7031  Basecbs 15637  s cress 15638  Hom chom 15721  Catccat 16090  Homf chomf 16092  compfccomf 16093  cat cresc 16233  Subcatcsubc 16234   Func cfunc 16279  idfunccidfu 16280   Full cful 16327   Faith cfth 16328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-er 7602  df-map 7719  df-pm 7720  df-ixp 7768  df-en 7815  df-dom 7816  df-sdom 7817  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-2 10922  df-3 10923  df-4 10924  df-5 10925  df-6 10926  df-7 10927  df-8 10928  df-9 10929  df-n0 11136  df-z 11207  df-dec 11322  df-ndx 15640  df-slot 15641  df-base 15642  df-sets 15643  df-ress 15644  df-hom 15735  df-cco 15736  df-cat 16094  df-cid 16095  df-homf 16096  df-comf 16097  df-ssc 16235  df-resc 16236  df-subc 16237  df-func 16283  df-idfu 16284  df-full 16329  df-fth 16330
This theorem is referenced by: (None)
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