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Theorem fullresc 17902
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 2735 . . . . . 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 3996 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥𝐵)
8 simprr 773 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑦𝑆)
95, 8sseldd 3996 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑦𝐵)
101, 2, 3, 7, 9homfval 17737 . . . . 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 17738 . . . . . . . . 9 𝐻 Fn (𝐵 × 𝐵)
15 xpss12 5704 . . . . . . . . . 10 ((𝑆𝐵𝑆𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
164, 4, 15syl2anc 584 . . . . . . . . 9 (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
17 fnssres 6692 . . . . . . . . 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 17879 . . . . . . 7 (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) = (Hom ‘𝐸))
2019oveqdr 7459 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐸)𝑦))
2111, 20eqtr3d 2777 . . . . 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 6920 . . . . . . . 8 (Base‘𝐷) ∈ V
2624, 25eqeltrdi 2847 . . . . . . 7 (𝜑𝑆 ∈ V)
2722, 3resshom 17465 . . . . . . 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 2784 . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
3130ralrimivva 3200 . . 3 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
32 eqid 2735 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
33 eqid 2735 . . . 4 (Hom ‘𝐸) = (Hom ‘𝐸)
3412, 2, 13, 18, 4rescbas 17877 . . . 4 (𝜑𝑆 = (Base‘𝐸))
3532, 33, 24, 34homfeq 17739 . . 3 (𝜑 → ((Homf𝐷) = (Homf𝐸) ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦)))
3631, 35mpbird 257 . 2 (𝜑 → (Homf𝐷) = (Homf𝐸))
37 eqid 2735 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
3822, 37ressco 17466 . . . . 5 (𝑆 ∈ V → (comp‘𝐶) = (comp‘𝐷))
3926, 38syl 17 . . . 4 (𝜑 → (comp‘𝐶) = (comp‘𝐷))
4012, 2, 13, 18, 4, 37rescco 17881 . . . 4 (𝜑 → (comp‘𝐶) = (comp‘𝐸))
4139, 40eqtr3d 2777 . . 3 (𝜑 → (comp‘𝐷) = (comp‘𝐸))
4241, 36comfeqd 17752 . 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 1537  wcel 2106  wral 3059  Vcvv 3478  wss 3963   × cxp 5687  cres 5691   Fn wfn 6558  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  Hom chom 17309  compcco 17310  Catccat 17709  Homf chomf 17711  compfccomf 17712  cat cresc 17856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-hom 17322  df-cco 17323  df-homf 17715  df-comf 17716  df-resc 17859
This theorem is referenced by:  resscat  17903  funcres2c  17955  ressffth  17992  funcsetcres2  18147
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