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Theorem fvssunirn 6925
Description: The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 31-Aug-2015.) (Proof shortened by SN, 13-Jan-2025.)
Assertion
Ref Expression
fvssunirn (𝐹𝑋) ⊆ ran 𝐹

Proof of Theorem fvssunirn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elfvunirn 6924 . 2 (𝑥 ∈ (𝐹𝑋) → 𝑥 ran 𝐹)
21ssriv 3987 1 (𝐹𝑋) ⊆ ran 𝐹
Colors of variables: wff setvar class
Syntax hints:  wss 3949   cuni 4909  ran crn 5678  cfv 6544
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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-cnv 5685  df-dm 5687  df-rn 5688  df-iota 6496  df-fv 6552
This theorem is referenced by:  ovssunirn  7445  marypha2lem1  9430  acnlem  10043  fin23lem29  10336  itunitc  10416  hsmexlem5  10425  wunfv  10727  wunex2  10733  strfvss  17120  prdsvallem  17400  prdsval  17401  prdsbas  17403  prdsplusg  17404  prdsmulr  17405  prdsvsca  17406  prdshom  17413  mreunirn  17545  mrcfval  17552  mrcssv  17558  mrisval  17574  sscpwex  17762  wunfunc  17849  wunfuncOLD  17850  catcxpccl  18159  catcxpcclOLD  18160  comppfsc  23036  filunirn  23386  elflim  23475  flffval  23493  fclsval  23512  isfcls  23513  fcfval  23537  tsmsxplem1  23657  xmetunirn  23843  mopnval  23944  tmsval  23989  cfilfval  24781  caufval  24792  issgon  33121  elrnsiga  33124  volmeas  33229  omssubadd  33299  neibastop2lem  35245  ctbssinf  36287  ismtyval  36668  dicval  40047  prjcrv0  41375  ismrc  41439  nacsfix  41450  hbt  41872
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