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Theorem fullresc 17896
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 2737 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
4 fullsubc.s . . . . . . . 8 (𝜑𝑆𝐵)
54adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑆𝐵)
6 simprl 771 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥𝑆)
75, 6sseldd 3984 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥𝐵)
8 simprr 773 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑦𝑆)
95, 8sseldd 3984 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑦𝐵)
101, 2, 3, 7, 9homfval 17735 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
116, 8ovresd 7600 . . . . . 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 17736 . . . . . . . . 9 𝐻 Fn (𝐵 × 𝐵)
15 xpss12 5700 . . . . . . . . . 10 ((𝑆𝐵𝑆𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
164, 4, 15syl2anc 584 . . . . . . . . 9 (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
17 fnssres 6691 . . . . . . . . 9 ((𝐻 Fn (𝐵 × 𝐵) ∧ (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
1814, 16, 17sylancr 587 . . . . . . . 8 (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
1912, 2, 13, 18, 4reschom 17874 . . . . . . 7 (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) = (Hom ‘𝐸))
2019oveqdr 7459 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐸)𝑦))
2111, 20eqtr3d 2779 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐸)𝑦))
22 fullsubc.d . . . . . . . . . 10 𝐷 = (𝐶s 𝑆)
2322, 2ressbas2 17283 . . . . . . . . 9 (𝑆𝐵𝑆 = (Base‘𝐷))
244, 23syl 17 . . . . . . . 8 (𝜑𝑆 = (Base‘𝐷))
25 fvex 6919 . . . . . . . 8 (Base‘𝐷) ∈ V
2624, 25eqeltrdi 2849 . . . . . . 7 (𝜑𝑆 ∈ V)
2722, 3resshom 17463 . . . . . . 7 (𝑆 ∈ V → (Hom ‘𝐶) = (Hom ‘𝐷))
2826, 27syl 17 . . . . . 6 (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷))
2928oveqdr 7459 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
3010, 21, 293eqtr3rd 2786 . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
3130ralrimivva 3202 . . 3 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
32 eqid 2737 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
33 eqid 2737 . . . 4 (Hom ‘𝐸) = (Hom ‘𝐸)
3412, 2, 13, 18, 4rescbas 17873 . . . 4 (𝜑𝑆 = (Base‘𝐸))
3532, 33, 24, 34homfeq 17737 . . 3 (𝜑 → ((Homf𝐷) = (Homf𝐸) ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦)))
3631, 35mpbird 257 . 2 (𝜑 → (Homf𝐷) = (Homf𝐸))
37 eqid 2737 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
3822, 37ressco 17464 . . . . 5 (𝑆 ∈ V → (comp‘𝐶) = (comp‘𝐷))
3926, 38syl 17 . . . 4 (𝜑 → (comp‘𝐶) = (comp‘𝐷))
4012, 2, 13, 18, 4, 37rescco 17876 . . . 4 (𝜑 → (comp‘𝐶) = (comp‘𝐸))
4139, 40eqtr3d 2779 . . 3 (𝜑 → (comp‘𝐷) = (comp‘𝐸))
4241, 36comfeqd 17750 . 2 (𝜑 → (compf𝐷) = (compf𝐸))
4336, 42jca 511 1 (𝜑 → ((Homf𝐷) = (Homf𝐸) ∧ (compf𝐷) = (compf𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  wss 3951   × cxp 5683  cres 5687   Fn wfn 6556  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  Hom chom 17308  compcco 17309  Catccat 17707  Homf chomf 17709  compfccomf 17710  cat cresc 17852
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-hom 17321  df-cco 17322  df-homf 17713  df-comf 17714  df-resc 17855
This theorem is referenced by:  resscat  17897  funcres2c  17948  ressffth  17985  funcsetcres2  18138
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