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

Theorem ovssunirn 6637
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 6610 . 2 (𝑋𝐹𝑌) = (𝐹‘⟨𝑋, 𝑌⟩)
2 fvssunirn 6176 . 2 (𝐹‘⟨𝑋, 𝑌⟩) ⊆ ran 𝐹
31, 2eqsstri 3616 1 (𝑋𝐹𝑌) ⊆ ran 𝐹
Colors of variables: wff setvar class
Syntax hints:  wss 3556  cop 4156   cuni 4404  ran crn 5077  cfv 5849  (class class class)co 6607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-cnv 5084  df-dm 5086  df-rn 5087  df-iota 5812  df-fv 5857  df-ov 6610
This theorem is referenced by:  prdsval  16039  prdsplusg  16042  prdsmulr  16043  prdsvsca  16044  prdshom  16051  wunfunc  16483  wunnat  16540  homarw  16620  catcoppccl  16682  catcfuccl  16683  catcxpccl  16771
  Copyright terms: Public domain W3C validator