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Theorem fullresc 17124
Description: The category formed by structure restriction is the same as the category restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
Hypotheses
Ref Expression
fullsubc.b 𝐵 = (Base‘𝐶)
fullsubc.h 𝐻 = (Homf𝐶)
fullsubc.c (𝜑𝐶 ∈ Cat)
fullsubc.s (𝜑𝑆𝐵)
fullsubc.d 𝐷 = (𝐶s 𝑆)
fullsubc.e 𝐸 = (𝐶cat (𝐻 ↾ (𝑆 × 𝑆)))
Assertion
Ref Expression
fullresc (𝜑 → ((Homf𝐷) = (Homf𝐸) ∧ (compf𝐷) = (compf𝐸)))

Proof of Theorem fullresc
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fullsubc.h . . . . . 6 𝐻 = (Homf𝐶)
2 fullsubc.b . . . . . 6 𝐵 = (Base‘𝐶)
3 eqid 2824 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
4 fullsubc.s . . . . . . . 8 (𝜑𝑆𝐵)
54adantr 483 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑆𝐵)
6 simprl 769 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥𝑆)
75, 6sseldd 3971 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥𝐵)
8 simprr 771 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑦𝑆)
95, 8sseldd 3971 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑦𝐵)
101, 2, 3, 7, 9homfval 16965 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
116, 8ovresd 7318 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥𝐻𝑦))
12 fullsubc.e . . . . . . . 8 𝐸 = (𝐶cat (𝐻 ↾ (𝑆 × 𝑆)))
13 fullsubc.c . . . . . . . 8 (𝜑𝐶 ∈ Cat)
141, 2homffn 16966 . . . . . . . . 9 𝐻 Fn (𝐵 × 𝐵)
15 xpss12 5573 . . . . . . . . . 10 ((𝑆𝐵𝑆𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
164, 4, 15syl2anc 586 . . . . . . . . 9 (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
17 fnssres 6473 . . . . . . . . 9 ((𝐻 Fn (𝐵 × 𝐵) ∧ (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
1814, 16, 17sylancr 589 . . . . . . . 8 (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
1912, 2, 13, 18, 4reschom 17103 . . . . . . 7 (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) = (Hom ‘𝐸))
2019oveqdr 7187 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐸)𝑦))
2111, 20eqtr3d 2861 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐸)𝑦))
22 fullsubc.d . . . . . . . . . 10 𝐷 = (𝐶s 𝑆)
2322, 2ressbas2 16558 . . . . . . . . 9 (𝑆𝐵𝑆 = (Base‘𝐷))
244, 23syl 17 . . . . . . . 8 (𝜑𝑆 = (Base‘𝐷))
25 fvex 6686 . . . . . . . 8 (Base‘𝐷) ∈ V
2624, 25eqeltrdi 2924 . . . . . . 7 (𝜑𝑆 ∈ V)
2722, 3resshom 16694 . . . . . . 7 (𝑆 ∈ V → (Hom ‘𝐶) = (Hom ‘𝐷))
2826, 27syl 17 . . . . . 6 (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷))
2928oveqdr 7187 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
3010, 21, 293eqtr3rd 2868 . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
3130ralrimivva 3194 . . 3 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
32 eqid 2824 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
33 eqid 2824 . . . 4 (Hom ‘𝐸) = (Hom ‘𝐸)
3412, 2, 13, 18, 4rescbas 17102 . . . 4 (𝜑𝑆 = (Base‘𝐸))
3532, 33, 24, 34homfeq 16967 . . 3 (𝜑 → ((Homf𝐷) = (Homf𝐸) ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦)))
3631, 35mpbird 259 . 2 (𝜑 → (Homf𝐷) = (Homf𝐸))
37 eqid 2824 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
3822, 37ressco 16695 . . . . 5 (𝑆 ∈ V → (comp‘𝐶) = (comp‘𝐷))
3926, 38syl 17 . . . 4 (𝜑 → (comp‘𝐶) = (comp‘𝐷))
4012, 2, 13, 18, 4, 37rescco 17105 . . . 4 (𝜑 → (comp‘𝐶) = (comp‘𝐸))
4139, 40eqtr3d 2861 . . 3 (𝜑 → (comp‘𝐷) = (comp‘𝐸))
4241, 36comfeqd 16980 . 2 (𝜑 → (compf𝐷) = (compf𝐸))
4336, 42jca 514 1 (𝜑 → ((Homf𝐷) = (Homf𝐸) ∧ (compf𝐷) = (compf𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1536  wcel 2113  wral 3141  Vcvv 3497  wss 3939   × cxp 5556  cres 5560   Fn wfn 6353  cfv 6358  (class class class)co 7159  Basecbs 16486  s cress 16487  Hom chom 16579  compcco 16580  Catccat 16938  Homf chomf 16940  compfccomf 16941  cat cresc 17081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-er 8292  df-en 8513  df-dom 8514  df-sdom 8515  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-ndx 16489  df-slot 16490  df-base 16492  df-sets 16493  df-ress 16494  df-hom 16592  df-cco 16593  df-homf 16944  df-comf 16945  df-resc 17084
This theorem is referenced by:  resscat  17125  funcres2c  17174  ressffth  17211  funcsetcres2  17356
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