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

Theorem ovssunirn 7181
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 (𝑋𝐹𝑌) ⊆ ran 𝐹

Proof of Theorem ovssunirn
StepHypRef Expression
1 df-ov 7148 . 2 (𝑋𝐹𝑌) = (𝐹‘⟨𝑋, 𝑌⟩)
2 fvssunirn 6693 . 2 (𝐹‘⟨𝑋, 𝑌⟩) ⊆ ran 𝐹
31, 2eqsstri 4000 1 (𝑋𝐹𝑌) ⊆ ran 𝐹
Colors of variables: wff setvar class
Syntax hints:  wss 3935  cop 4565   cuni 4832  ran crn 5550  cfv 6349  (class class class)co 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-cnv 5557  df-dm 5559  df-rn 5560  df-iota 6308  df-fv 6357  df-ov 7148
This theorem is referenced by:  prdsval  16718  prdsplusg  16721  prdsmulr  16722  prdsvsca  16723  prdshom  16730  wunfunc  17159  wunnat  17216  homarw  17296  catcoppccl  17358  catcfuccl  17359  catcxpccl  17447
  Copyright terms: Public domain W3C validator