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

Theorem ovssunirn 7394
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 7361 . 2 (𝑋𝐹𝑌) = (𝐹‘⟨𝑋, 𝑌⟩)
2 fvssunirn 6876 . 2 (𝐹‘⟨𝑋, 𝑌⟩) ⊆ ran 𝐹
31, 2eqsstri 3979 1 (𝑋𝐹𝑌) ⊆ ran 𝐹
Colors of variables: wff setvar class
Syntax hints:  wss 3911  cop 4593   cuni 4866  ran crn 5635  cfv 6497  (class class class)co 7358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-cnv 5642  df-dm 5644  df-rn 5645  df-iota 6449  df-fv 6505  df-ov 7361
This theorem is referenced by:  prdsvallem  17337  prdsplusg  17341  prdsmulr  17342  prdsvsca  17343  prdshom  17350  wunfunc  17786  wunfuncOLD  17787  wunnat  17844  wunnatOLD  17845  homarw  17933  catcoppccl  18004  catcoppcclOLD  18005  catcfuccl  18006  catcfucclOLD  18007  catcxpccl  18096  catcxpcclOLD  18097  isanmbfmOLD  32857  isanmbfm  32859
  Copyright terms: Public domain W3C validator