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Theorem ressffth 16421
 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 16345 . . 3 Rel (𝐷 Func 𝐷)
2 ressffth.d . . . . 5 𝐷 = (𝐶s 𝑆)
3 resscat 16335 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐶s 𝑆) ∈ Cat)
42, 3syl5eqel 2692 . . . 4 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐷 ∈ Cat)
5 ressffth.i . . . . 5 𝐼 = (idfunc𝐷)
65idfucl 16364 . . . 4 (𝐷 ∈ Cat → 𝐼 ∈ (𝐷 Func 𝐷))
74, 6syl 17 . . 3 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 ∈ (𝐷 Func 𝐷))
8 1st2nd 7105 . . 3 ((Rel (𝐷 Func 𝐷) ∧ 𝐼 ∈ (𝐷 Func 𝐷)) → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
91, 7, 8sylancr 694 . 2 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
10 eqidd 2611 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf𝐷) = (Homf𝐷))
11 eqidd 2611 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf𝐷) = (compf𝐷))
12 eqid 2610 . . . . . . . . . . . . . 14 (Base‘𝐶) = (Base‘𝐶)
1312ressinbas 15763 . . . . . . . . . . . . 13 (𝑆𝑉 → (𝐶s 𝑆) = (𝐶s (𝑆 ∩ (Base‘𝐶))))
1413adantl 481 . . . . . . . . . . . 12 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐶s 𝑆) = (𝐶s (𝑆 ∩ (Base‘𝐶))))
152, 14syl5eq 2656 . . . . . . . . . . 11 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐷 = (𝐶s (𝑆 ∩ (Base‘𝐶))))
1615fveq2d 6107 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf𝐷) = (Homf ‘(𝐶s (𝑆 ∩ (Base‘𝐶)))))
17 eqid 2610 . . . . . . . . . . . 12 (Homf𝐶) = (Homf𝐶)
18 simpl 472 . . . . . . . . . . . 12 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐶 ∈ Cat)
19 inss2 3796 . . . . . . . . . . . . 13 (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶)
2019a1i 11 . . . . . . . . . . . 12 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶))
21 eqid 2610 . . . . . . . . . . . 12 (𝐶s (𝑆 ∩ (Base‘𝐶))) = (𝐶s (𝑆 ∩ (Base‘𝐶)))
22 eqid 2610 . . . . . . . . . . . 12 (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶))))) = (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))
2312, 17, 18, 20, 21, 22fullresc 16334 . . . . . . . . . . 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 474 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf ‘(𝐶s (𝑆 ∩ (Base‘𝐶)))) = (Homf ‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
2516, 24eqtrd 2644 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (Homf𝐷) = (Homf ‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
2615fveq2d 6107 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf𝐷) = (compf‘(𝐶s (𝑆 ∩ (Base‘𝐶)))))
2723simprd 478 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf‘(𝐶s (𝑆 ∩ (Base‘𝐶)))) = (compf‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
2826, 27eqtrd 2644 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (compf𝐷) = (compf‘(𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
29 ovex 6577 . . . . . . . . . . 11 (𝐶s 𝑆) ∈ V
302, 29eqeltri 2684 . . . . . . . . . 10 𝐷 ∈ V
3130a1i 11 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐷 ∈ V)
32 ovex 6577 . . . . . . . . . 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 16383 . . . . . . . 8 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐷 Func 𝐷) = (𝐷 Func (𝐶cat ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))))))
3512, 17, 18, 20fullsubc 16333 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → ((Homf𝐶) ↾ ((𝑆 ∩ (Base‘𝐶)) × (𝑆 ∩ (Base‘𝐶)))) ∈ (Subcat‘𝐶))
36 funcres2 16381 . . . . . . . . 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 3602 . . . . . . 7 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐷 Func 𝐷) ⊆ (𝐷 Func 𝐶))
3938, 7sseldd 3569 . . . . . 6 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 ∈ (𝐷 Func 𝐶))
409, 39eqeltrrd 2689 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → ⟨(1st𝐼), (2nd𝐼)⟩ ∈ (𝐷 Func 𝐶))
41 df-br 4584 . . . . 5 ((1st𝐼)(𝐷 Func 𝐶)(2nd𝐼) ↔ ⟨(1st𝐼), (2nd𝐼)⟩ ∈ (𝐷 Func 𝐶))
4240, 41sylibr 223 . . . 4 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (1st𝐼)(𝐷 Func 𝐶)(2nd𝐼))
43 f1oi 6086 . . . . . 6 ( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(𝑥(Hom ‘𝐷)𝑦)
44 eqid 2610 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
454adantr 480 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐷 ∈ Cat)
46 eqid 2610 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
47 simprl 790 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
48 simprr 792 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
495, 44, 45, 46, 47, 48idfu2nd 16360 . . . . . . 7 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(2nd𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
50 eqidd 2611 . . . . . . 7 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
51 eqid 2610 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
522, 51resshom 15901 . . . . . . . . 9 (𝑆𝑉 → (Hom ‘𝐶) = (Hom ‘𝐷))
5352ad2antlr 759 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Hom ‘𝐶) = (Hom ‘𝐷))
545, 44, 45, 47idfu1 16363 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st𝐼)‘𝑥) = 𝑥)
555, 44, 45, 48idfu1 16363 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st𝐼)‘𝑦) = 𝑦)
5653, 54, 55oveq123d 6570 . . . . . . 7 (((𝐶 ∈ Cat ∧ 𝑆𝑉) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (((1st𝐼)‘𝑥)(Hom ‘𝐶)((1st𝐼)‘𝑦)) = (𝑥(Hom ‘𝐷)𝑦))
5749, 50, 56f1oeq123d 6046 . . . . . 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 247 . . . . 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 2954 . . . 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 16399 . . . 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 695 . . 3 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → (1st𝐼)((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶))(2nd𝐼))
62 df-br 4584 . . 3 ((1st𝐼)((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶))(2nd𝐼) ↔ ⟨(1st𝐼), (2nd𝐼)⟩ ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
6361, 62sylib 207 . 2 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → ⟨(1st𝐼), (2nd𝐼)⟩ ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
649, 63eqeltrd 2688 1 ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ⟨cop 4131   class class class wbr 4583   I cid 4948   × cxp 5036   ↾ cres 5040  Rel wrel 5043  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Basecbs 15695   ↾s cress 15696  Hom chom 15779  Catccat 16148  Homf chomf 16150  compfccomf 16151   ↾cat cresc 16291  Subcatcsubc 16292   Func cfunc 16337  idfunccidfu 16338   Full cful 16385   Faith cfth 16386 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-hom 15793  df-cco 15794  df-cat 16152  df-cid 16153  df-homf 16154  df-comf 16155  df-ssc 16293  df-resc 16294  df-subc 16295  df-func 16341  df-idfu 16342  df-full 16387  df-fth 16388 This theorem is referenced by: (None)
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