MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subsubc Structured version   Unicode version

Theorem subsubc 15097
Description: A subcategory of a subcategory is a subcategory. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypothesis
Ref Expression
subsubc.d  |-  D  =  ( C  |`cat  H )
Assertion
Ref Expression
subsubc  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  <->  ( J  e.  (Subcat `  C )  /\  J  C_cat  H ) ) )

Proof of Theorem subsubc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . 6  |-  ( J  e.  (Subcat `  D
)  ->  J  e.  (Subcat `  D ) )
2 eqid 2467 . . . . . 6  |-  ( Hom f  `  D )  =  ( Hom f  `  D )
31, 2subcssc 15084 . . . . 5  |-  ( J  e.  (Subcat `  D
)  ->  J  C_cat  ( Hom f  `  D ) )
4 subsubc.d . . . . . . 7  |-  D  =  ( C  |`cat  H )
5 eqid 2467 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
6 subcrcl 15063 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  C  e.  Cat )
7 id 22 . . . . . . . 8  |-  ( H  e.  (Subcat `  C
)  ->  H  e.  (Subcat `  C ) )
8 eqidd 2468 . . . . . . . 8  |-  ( H  e.  (Subcat `  C
)  ->  dom  dom  H  =  dom  dom  H )
97, 8subcfn 15085 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  H  Fn  ( dom  dom  H  X.  dom  dom  H ) )
107, 9, 5subcss1 15086 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  dom  dom  H  C_  ( Base `  C
) )
114, 5, 6, 9, 10reschomf 15078 . . . . . 6  |-  ( H  e.  (Subcat `  C
)  ->  H  =  ( Hom f  `  D ) )
1211breq2d 4465 . . . . 5  |-  ( H  e.  (Subcat `  C
)  ->  ( J  C_cat  H  <-> 
J  C_cat  ( Hom f  `  D ) ) )
133, 12syl5ibr 221 . . . 4  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  ->  J  C_cat  H ) )
1413pm4.71rd 635 . . 3  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  <->  ( J  C_cat  H  /\  J  e.  (Subcat `  D )
) ) )
15 simpr 461 . . . . . . . 8  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  C_cat  H )
16 simpl 457 . . . . . . . . 9  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  H  e.  (Subcat `  C ) )
17 eqid 2467 . . . . . . . . 9  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
1816, 17subcssc 15084 . . . . . . . 8  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  H  C_cat  ( Hom f  `  C ) )
19 ssctr 15072 . . . . . . . 8  |-  ( ( J  C_cat  H  /\  H  C_cat  ( Hom f  `  C ) )  ->  J  C_cat  ( Hom f  `  C ) )
2015, 18, 19syl2anc 661 . . . . . . 7  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  C_cat  ( Hom f  `  C ) )
2112biimpa 484 . . . . . . 7  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  C_cat  ( Hom f  `  D ) )
2220, 212thd 240 . . . . . 6  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( J  C_cat  ( Hom f  `  C )  <->  J  C_cat  ( Hom f  `  D ) ) )
2316adantr 465 . . . . . . . . 9  |-  ( ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  /\  x  e. 
dom  dom  J )  ->  H  e.  (Subcat `  C
) )
249adantr 465 . . . . . . . . . 10  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  H  Fn  ( dom  dom  H  X.  dom  dom  H ) )
2524adantr 465 . . . . . . . . 9  |-  ( ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  /\  x  e. 
dom  dom  J )  ->  H  Fn  ( dom  dom 
H  X.  dom  dom  H ) )
26 eqid 2467 . . . . . . . . 9  |-  ( Id
`  C )  =  ( Id `  C
)
27 eqidd 2468 . . . . . . . . . . . 12  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  dom  dom  J  =  dom  dom  J )
2815, 27sscfn1 15064 . . . . . . . . . . 11  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  Fn  ( dom  dom  J  X.  dom  dom  J ) )
2928, 24, 15ssc1 15068 . . . . . . . . . 10  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  dom  dom  J  C_ 
dom  dom  H )
3029sselda 3509 . . . . . . . . 9  |-  ( ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  /\  x  e. 
dom  dom  J )  ->  x  e.  dom  dom  H
)
314, 23, 25, 26, 30subcid 15091 . . . . . . . 8  |-  ( ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  /\  x  e. 
dom  dom  J )  -> 
( ( Id `  C ) `  x
)  =  ( ( Id `  D ) `
 x ) )
3231eleq1d 2536 . . . . . . 7  |-  ( ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  /\  x  e. 
dom  dom  J )  -> 
( ( ( Id
`  C ) `  x )  e.  ( x J x )  <-> 
( ( Id `  D ) `  x
)  e.  ( x J x ) ) )
3332ralbidva 2903 . . . . . 6  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( A. x  e.  dom  dom  J
( ( Id `  C ) `  x
)  e.  ( x J x )  <->  A. x  e.  dom  dom  J (
( Id `  D
) `  x )  e.  ( x J x ) ) )
344oveq1i 6305 . . . . . . . 8  |-  ( D  |`cat 
J )  =  ( ( C  |`cat  H )  |`cat  J )
356adantr 465 . . . . . . . . 9  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  C  e.  Cat )
36 dmexg 6726 . . . . . . . . . . 11  |-  ( H  e.  (Subcat `  C
)  ->  dom  H  e. 
_V )
37 dmexg 6726 . . . . . . . . . . 11  |-  ( dom 
H  e.  _V  ->  dom 
dom  H  e.  _V )
3836, 37syl 16 . . . . . . . . . 10  |-  ( H  e.  (Subcat `  C
)  ->  dom  dom  H  e.  _V )
3938adantr 465 . . . . . . . . 9  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  dom  dom  H  e.  _V )
4035, 24, 28, 39, 29rescabs 15080 . . . . . . . 8  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( ( C  |`cat  H )  |`cat  J )  =  ( C  |`cat  J
) )
4134, 40syl5req 2521 . . . . . . 7  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( C  |`cat  J )  =  ( D  |`cat 
J ) )
4241eleq1d 2536 . . . . . 6  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( ( C  |`cat  J )  e.  Cat  <->  ( D  |`cat  J )  e.  Cat ) )
4322, 33, 423anbi123d 1299 . . . . 5  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( ( J  C_cat  ( Hom f  `  C )  /\  A. x  e.  dom  dom  J ( ( Id `  C ) `  x
)  e.  ( x J x )  /\  ( C  |`cat  J )  e.  Cat ) 
<->  ( J  C_cat  ( Hom f  `  D )  /\  A. x  e.  dom  dom  J
( ( Id `  D ) `  x
)  e.  ( x J x )  /\  ( D  |`cat  J )  e.  Cat ) ) )
44 eqid 2467 . . . . . 6  |-  ( C  |`cat 
J )  =  ( C  |`cat  J )
4517, 26, 44, 35, 28issubc3 15093 . . . . 5  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( J  e.  (Subcat `  C )  <->  ( J  C_cat  ( Hom f  `  C )  /\  A. x  e.  dom  dom  J ( ( Id `  C ) `  x
)  e.  ( x J x )  /\  ( C  |`cat  J )  e.  Cat ) ) )
46 eqid 2467 . . . . . 6  |-  ( Id
`  D )  =  ( Id `  D
)
47 eqid 2467 . . . . . 6  |-  ( D  |`cat 
J )  =  ( D  |`cat  J )
484, 7subccat 15092 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  D  e.  Cat )
4948adantr 465 . . . . . 6  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  D  e.  Cat )
502, 46, 47, 49, 28issubc3 15093 . . . . 5  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( J  e.  (Subcat `  D )  <->  ( J  C_cat  ( Hom f  `  D )  /\  A. x  e.  dom  dom  J ( ( Id `  D ) `  x
)  e.  ( x J x )  /\  ( D  |`cat  J )  e.  Cat ) ) )
5143, 45, 503bitr4rd 286 . . . 4  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( J  e.  (Subcat `  D )  <->  J  e.  (Subcat `  C
) ) )
5251pm5.32da 641 . . 3  |-  ( H  e.  (Subcat `  C
)  ->  ( ( J  C_cat  H  /\  J  e.  (Subcat `  D )
)  <->  ( J  C_cat  H  /\  J  e.  (Subcat `  C ) ) ) )
5314, 52bitrd 253 . 2  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  <->  ( J  C_cat  H  /\  J  e.  (Subcat `  C )
) ) )
54 ancom 450 . 2  |-  ( ( J  C_cat  H  /\  J  e.  (Subcat `  C )
)  <->  ( J  e.  (Subcat `  C )  /\  J  C_cat  H )
)
5553, 54syl6bb 261 1  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  <->  ( J  e.  (Subcat `  C )  /\  J  C_cat  H ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118   class class class wbr 4453    X. cxp 5003   dom cdm 5005    Fn wfn 5589   ` cfv 5594  (class class class)co 6295   Basecbs 14507   Catccat 14936   Idccid 14937   Hom f chomf 14938    C_cat cssc 15054    |`cat cresc 15055  Subcatcsubc 15056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-hom 14596  df-cco 14597  df-cat 14940  df-cid 14941  df-homf 14942  df-ssc 15057  df-resc 15058  df-subc 15059
This theorem is referenced by:  fldhmsubc  32397
  Copyright terms: Public domain W3C validator