MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subcss1 Unicode version

Theorem subcss1 13732
Description: The objects of a subcategory are a subset of the objects of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
subcss1.1  |-  ( ph  ->  J  e.  (Subcat `  C ) )
subcss1.2  |-  ( ph  ->  J  Fn  ( S  X.  S ) )
subcss1.b  |-  B  =  ( Base `  C
)
Assertion
Ref Expression
subcss1  |-  ( ph  ->  S  C_  B )

Proof of Theorem subcss1
StepHypRef Expression
1 subcss1.2 . 2  |-  ( ph  ->  J  Fn  ( S  X.  S ) )
2 eqid 2296 . . . 4  |-  (  Homf  `  C )  =  (  Homf 
`  C )
3 subcss1.b . . . 4  |-  B  =  ( Base `  C
)
42, 3homffn 13612 . . 3  |-  (  Homf  `  C )  Fn  ( B  X.  B )
54a1i 10 . 2  |-  ( ph  ->  (  Homf 
`  C )  Fn  ( B  X.  B
) )
6 subcss1.1 . . 3  |-  ( ph  ->  J  e.  (Subcat `  C ) )
76, 2subcssc 13730 . 2  |-  ( ph  ->  J  C_cat  (  Homf 
`  C ) )
81, 5, 7ssc1 13714 1  |-  ( ph  ->  S  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    C_ wss 3165    X. cxp 4703    Fn wfn 5266   ` cfv 5271   Basecbs 13164    Homf chomf 13584  Subcatcsubc 13702
This theorem is referenced by:  subcss2  13733  subccatid  13736  subsubc  13743  funcres  13786  funcres2b  13787  funcres2  13788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-pm 6791  df-ixp 6834  df-homf 13588  df-ssc 13703  df-subc 13705
  Copyright terms: Public domain W3C validator