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

Theorem fullresc 16864
 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 2826 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
4 fullsubc.s . . . . . . . 8 (𝜑𝑆𝐵)
54adantr 474 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑆𝐵)
6 simprl 789 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥𝑆)
75, 6sseldd 3829 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥𝐵)
8 simprr 791 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑦𝑆)
95, 8sseldd 3829 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑦𝐵)
101, 2, 3, 7, 9homfval 16705 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
116, 8ovresd 7062 . . . . . 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 16706 . . . . . . . . 9 𝐻 Fn (𝐵 × 𝐵)
15 xpss12 5358 . . . . . . . . . 10 ((𝑆𝐵𝑆𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
164, 4, 15syl2anc 581 . . . . . . . . 9 (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
17 fnssres 6238 . . . . . . . . 9 ((𝐻 Fn (𝐵 × 𝐵) ∧ (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
1814, 16, 17sylancr 583 . . . . . . . 8 (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
1912, 2, 13, 18, 4reschom 16843 . . . . . . 7 (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) = (Hom ‘𝐸))
2019oveqdr 6934 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐸)𝑦))
2111, 20eqtr3d 2864 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐸)𝑦))
22 fullsubc.d . . . . . . . . . 10 𝐷 = (𝐶s 𝑆)
2322, 2ressbas2 16295 . . . . . . . . 9 (𝑆𝐵𝑆 = (Base‘𝐷))
244, 23syl 17 . . . . . . . 8 (𝜑𝑆 = (Base‘𝐷))
25 fvex 6447 . . . . . . . 8 (Base‘𝐷) ∈ V
2624, 25syl6eqel 2915 . . . . . . 7 (𝜑𝑆 ∈ V)
2722, 3resshom 16432 . . . . . . 7 (𝑆 ∈ V → (Hom ‘𝐶) = (Hom ‘𝐷))
2826, 27syl 17 . . . . . 6 (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷))
2928oveqdr 6934 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
3010, 21, 293eqtr3rd 2871 . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
3130ralrimivva 3181 . . 3 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
32 eqid 2826 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
33 eqid 2826 . . . 4 (Hom ‘𝐸) = (Hom ‘𝐸)
3412, 2, 13, 18, 4rescbas 16842 . . . 4 (𝜑𝑆 = (Base‘𝐸))
3532, 33, 24, 34homfeq 16707 . . 3 (𝜑 → ((Homf𝐷) = (Homf𝐸) ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦)))
3631, 35mpbird 249 . 2 (𝜑 → (Homf𝐷) = (Homf𝐸))
37 eqid 2826 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
3822, 37ressco 16433 . . . . 5 (𝑆 ∈ V → (comp‘𝐶) = (comp‘𝐷))
3926, 38syl 17 . . . 4 (𝜑 → (comp‘𝐶) = (comp‘𝐷))
4012, 2, 13, 18, 4, 37rescco 16845 . . . 4 (𝜑 → (comp‘𝐶) = (comp‘𝐸))
4139, 40eqtr3d 2864 . . 3 (𝜑 → (comp‘𝐷) = (comp‘𝐸))
4241, 36comfeqd 16720 . 2 (𝜑 → (compf𝐷) = (compf𝐸))
4336, 42jca 509 1 (𝜑 → ((Homf𝐷) = (Homf𝐸) ∧ (compf𝐷) = (compf𝐸)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∀wral 3118  Vcvv 3415   ⊆ wss 3799   × cxp 5341   ↾ cres 5345   Fn wfn 6119  ‘cfv 6124  (class class class)co 6906  Basecbs 16223   ↾s cress 16224  Hom chom 16317  compcco 16318  Catccat 16678  Homf chomf 16680  compfccomf 16681   ↾cat cresc 16821 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-resscn 10310  ax-1cn 10311  ax-icn 10312  ax-addcl 10313  ax-addrcl 10314  ax-mulcl 10315  ax-mulrcl 10316  ax-mulcom 10317  ax-addass 10318  ax-mulass 10319  ax-distr 10320  ax-i2m1 10321  ax-1ne0 10322  ax-1rid 10323  ax-rnegex 10324  ax-rrecex 10325  ax-cnre 10326  ax-pre-lttri 10327  ax-pre-lttrn 10328  ax-pre-ltadd 10329  ax-pre-mulgt0 10330 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-nel 3104  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-1st 7429  df-2nd 7430  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-er 8010  df-en 8224  df-dom 8225  df-sdom 8226  df-pnf 10394  df-mnf 10395  df-xr 10396  df-ltxr 10397  df-le 10398  df-sub 10588  df-neg 10589  df-nn 11352  df-2 11415  df-3 11416  df-4 11417  df-5 11418  df-6 11419  df-7 11420  df-8 11421  df-9 11422  df-n0 11620  df-z 11706  df-dec 11823  df-ndx 16226  df-slot 16227  df-base 16229  df-sets 16230  df-ress 16231  df-hom 16330  df-cco 16331  df-homf 16684  df-comf 16685  df-resc 16824 This theorem is referenced by:  resscat  16865  funcres2c  16914  ressffth  16951  funcsetcres2  17096
 Copyright terms: Public domain W3C validator