Theorem fvssunirn 6900

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	⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹

Proof of Theorem fvssunirn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvunirn 6899	. 2 ⊢ (𝑥 ∈ (𝐹‘𝑋) → 𝑥 ∈ ∪ ran 𝐹)
2	1	ssriv 3942	1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹

Syntax hints:   ⊆ wss 3906  ∪ cuni 4867  ran crn 5650  ‘cfv 6523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-cnv 5657  df-dm 5659  df-rn 5660  df-iota 6479  df-fv 6531

This theorem is referenced by:  ovssunirn  7434  marypha2lem1  9383  acnlem  10006  fin23lem29  10300  itunitc  10380  hsmexlem5  10389  wunfv  10692  wunex2  10698  strfvss  17225  prdsvallem  17485  prdsval  17486  prdsbas  17488  prdsplusg  17489  prdsmulr  17490  prdsvsca  17491  prdshom  17498  mreunirn  17631  mrcfval  17642  mrcssv  17648  mrisval  17664  sscpwex  17850  wunfunc  17936  catcxpccl  18241  comppfsc  23594  filunirn  23944  elflim  24033  flffval  24051  fclsval  24070  isfcls  24071  fcfval  24095  tsmsxplem1  24215  xmetunirn  24399  mopnval  24500  tmsval  24543  cfilfval  25328  caufval  25339  issgon  34422  elrnsiga  34425  volmeas  34530  omssubadd  34599  neibastop2lem  36725  ctbssinf  37905  ismtyval  38304  dicval  41805  prjcrv0  43220  ismrc  43287  nacsfix  43298  hbt  43712