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

Theorem ovssunirn 6129
Description: The result of an operation value is always a subset of the union of the range. (Contributed by Mario Carneiro, 12-Jan-2017.)
Assertion
Ref Expression
ovssunirn  |-  ( X F Y )  C_  U.
ran  F

Proof of Theorem ovssunirn
StepHypRef Expression
1 df-ov 6106 . 2  |-  ( X F Y )  =  ( F `  <. X ,  Y >. )
2 fvssunirn 5725 . 2  |-  ( F `
 <. X ,  Y >. )  C_  U. ran  F
31, 2eqsstri 3398 1  |-  ( X F Y )  C_  U.
ran  F
Colors of variables: wff setvar class
Syntax hints:    C_ wss 3340   <.cop 3895   U.cuni 4103   ran crn 4853   ` cfv 5430  (class class class)co 6103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-cnv 4860  df-dm 4862  df-rn 4863  df-iota 5393  df-fv 5438  df-ov 6106
This theorem is referenced by:  prdsval  14405  prdsplusg  14408  prdsmulr  14409  prdsvsca  14410  prdshom  14417  wunfunc  14821  wunnat  14878  homarw  14926  catcoppccl  14988  catcfuccl  14989  catcxpccl  15029
  Copyright terms: Public domain W3C validator