Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ressffth Structured version   Visualization version   GIF version

Theorem ressffth 17200
 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 17124 . . 3 Rel (𝐷 Func 𝐷)
2 ressffth.d . . . . 5 𝐷 = (𝐶s 𝑆)
3 resscat 17114 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐶s 𝑆) ∈ Cat)
42, 3eqeltrid 2915 . . . 4 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐷 ∈ Cat)
5 ressffth.i . . . . 5 𝐼 = (idfunc𝐷)
65idfucl 17143 . . . 4 (𝐷 ∈ Cat → 𝐼 ∈ (𝐷 Func 𝐷))
74, 6syl 17 . . 3 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 ∈ (𝐷 Func 𝐷))
8 1st2nd 7730 . . 3 ((Rel (𝐷 Func 𝐷) ∧ 𝐼 ∈ (𝐷 Func 𝐷)) → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
91, 7, 8sylancr 589 . 2 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
10 eqidd 2820 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf𝐷) = (Homf𝐷))
11 eqidd 2820 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf𝐷) = (compf𝐷))
12 eqid 2819 . . . . . . . . . . . . . 14 (Base‘𝐶) = (Base‘𝐶)
1312ressinbas 16552 . . . . . . . . . . . . 13 (𝑆𝑉 → (𝐶s 𝑆) = (𝐶s (𝑆 ∩ (Base‘𝐶))))
1413adantl 484 . . . . . . . . . . . 12 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐶s 𝑆) = (𝐶s (𝑆 ∩ (Base‘𝐶))))
152, 14syl5eq 2866 . . . . . . . . . . 11 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐷 = (𝐶s (𝑆 ∩ (Base‘𝐶))))
1615fveq2d 6667 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf𝐷) = (Homf ‘(𝐶s (𝑆 ∩ (Base‘𝐶)))))
17 eqid 2819 . . . . . . . . . . . 12 (Homf𝐶) = (Homf𝐶)
18 simpl 485 . . . . . . . . . . . 12 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐶 ∈ Cat)
19 inss2 4204 . . . . . . . . . . . . 13 (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶)
2019a1i 11 . . . . . . . . . . . 12 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶))
21 eqid 2819 . . . . . . . . . . . 12 (𝐶s (𝑆 ∩ (Base‘𝐶))) = (𝐶s (𝑆 ∩ (Base‘𝐶)))
22 eqid 2819 . . . . . . . . . . . 12 (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶))))) = (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))
2312, 17, 18, 20, 21, 22fullresc 17113 . . . . . . . . . . 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 497 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf ‘(𝐶s (𝑆 ∩ (Base‘𝐶)))) = (Homf ‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
2516, 24eqtrd 2854 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf𝐷) = (Homf ‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
2615fveq2d 6667 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf𝐷) = (compf‘(𝐶s (𝑆 ∩ (Base‘𝐶)))))
2723simprd 498 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf‘(𝐶s (𝑆 ∩ (Base‘𝐶)))) = (compf‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
2826, 27eqtrd 2854 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf𝐷) = (compf‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
292ovexi 7182 . . . . . . . . . 10 𝐷 ∈ V
3029a1i 11 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐷 ∈ V)
31 ovexd 7183 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶))))) ∈ V)
3210, 11, 25, 28, 30, 30, 30, 31funcpropd 17162 . . . . . . . 8 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐷 Func 𝐷) = (𝐷 Func (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
3312, 17, 18, 20fullsubc 17112 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))) ∈ (Subcat‘𝐶))
34 funcres2 17160 . . . . . . . . 9 (((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))) ∈ (Subcat‘𝐶) → (𝐷 Func (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))) ⊆ (𝐷 Func 𝐶))
3533, 34syl 17 . . . . . . . 8 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐷 Func (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))) ⊆ (𝐷 Func 𝐶))
3632, 35eqsstrd 4003 . . . . . . 7 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐷 Func 𝐷) ⊆ (𝐷 Func 𝐶))
3736, 7sseldd 3966 . . . . . 6 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 ∈ (𝐷 Func 𝐶))
389, 37eqeltrrd 2912 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → ⟨(1st𝐼), (2nd𝐼)⟩ ∈ (𝐷 Func 𝐶))
39 df-br 5058 . . . . 5 ((1st𝐼)(𝐷 Func 𝐶)(2nd𝐼) ↔ ⟨(1st𝐼), (2nd𝐼)⟩ ∈ (𝐷 Func 𝐶))
4038, 39sylibr 236 . . . 4 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (1st𝐼)(𝐷 Func 𝐶)(2nd𝐼))
41 f1oi 6645 . . . . . 6 ( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(𝑥(Hom ‘𝐷)𝑦)
42 eqid 2819 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
434adantr 483 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐷 ∈ Cat)
44 eqid 2819 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
45 simprl 769 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
46 simprr 771 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
475, 42, 43, 44, 45, 46idfu2nd 17139 . . . . . . 7 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(2nd𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
48 eqidd 2820 . . . . . . 7 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
49 eqid 2819 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
502, 49resshom 16683 . . . . . . . . 9 (𝑆𝑉 → (Hom ‘𝐶) = (Hom ‘𝐷))
5150ad2antlr 725 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Hom ‘𝐶) = (Hom ‘𝐷))
525, 42, 43, 45idfu1 17142 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st𝐼)‘𝑥) = 𝑥)
535, 42, 43, 46idfu1 17142 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st𝐼)‘𝑦) = 𝑦)
5451, 52, 53oveq123d 7169 . . . . . . 7 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (((1st𝐼)‘𝑥)(Hom ‘𝐶)((1st𝐼)‘𝑦)) = (𝑥(Hom ‘𝐷)𝑦))
5547, 48, 54f1oeq123d 6603 . . . . . 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)))
5641, 55mpbiri 260 . . . . 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)))
5756ralrimivva 3189 . . . 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)))
5842, 44, 49isffth2 17178 . . . 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))))
5940, 57, 58sylanbrc 585 . . 3 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (1st𝐼)((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶))(2nd𝐼))
60 df-br 5058 . . 3 ((1st𝐼)((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶))(2nd𝐼) ↔ ⟨(1st𝐼), (2nd𝐼)⟩ ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
6159, 60sylib 220 . 2 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → ⟨(1st𝐼), (2nd𝐼)⟩ ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
629, 61eqeltrd 2911 1 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1531   ∈ wcel 2108  ∀wral 3136  Vcvv 3493   ∩ cin 3933   ⊆ wss 3934  ⟨cop 4565   class class class wbr 5057   I cid 5452   × cxp 5546   ↾ cres 5550  Rel wrel 5553  –1-1-onto→wf1o 6347  ‘cfv 6348  (class class class)co 7148  1st c1st 7679  2nd c2nd 7680  Basecbs 16475   ↾s cress 16476  Hom chom 16568  Catccat 16927  Homf chomf 16929  compfccomf 16930   ↾cat cresc 17070  Subcatcsubc 17071   Func cfunc 17116  idfunccidfu 17117   Full cful 17164   Faith cfth 17165 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-fal 1544  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-er 8281  df-map 8400  df-pm 8401  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-hom 16581  df-cco 16582  df-cat 16931  df-cid 16932  df-homf 16933  df-comf 16934  df-ssc 17072  df-resc 17073  df-subc 17074  df-func 17120  df-idfu 17121  df-full 17166  df-fth 17167 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator