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

Theorem homarw 13894
Description: A hom-set is a subset of the collection of all arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
arwrcl.a  |-  A  =  (Nat `  C )
arwhoma.h  |-  H  =  (Homa
`  C )
Assertion
Ref Expression
homarw  |-  ( X H Y )  C_  A

Proof of Theorem homarw
StepHypRef Expression
1 ovssunirn 5900 . 2  |-  ( X H Y )  C_  U.
ran  H
2 arwrcl.a . . 3  |-  A  =  (Nat `  C )
3 arwhoma.h . . 3  |-  H  =  (Homa
`  C )
42, 3arwval 13891 . 2  |-  A  = 
U. ran  H
51, 4sseqtr4i 3224 1  |-  ( X H Y )  C_  A
Colors of variables: wff set class
Syntax hints:    = wceq 1632    C_ wss 3165   U.cuni 3843   ran crn 4706   ` cfv 5271  (class class class)co 5874  Natcarw 13870  Homachoma 13871
This theorem is referenced by:  idaf  13911  homdmcoa  13915  coaval  13916  coapm  13919
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-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-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  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-fv 5279  df-ov 5877  df-homa 13874  df-arw 13875
  Copyright terms: Public domain W3C validator