Theorem fvssunirn 6871

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 6870	. 2 ⊢ (𝑥 ∈ (𝐹‘𝑋) → 𝑥 ∈ ∪ ran 𝐹)
2	1	ssriv 3925	1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹

Syntax hints:   ⊆ wss 3889  ∪ cuni 4850  ran crn 5632  ‘cfv 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-cnv 5639  df-dm 5641  df-rn 5642  df-iota 6454  df-fv 6506

This theorem is referenced by:  ovssunirn  7403  marypha2lem1  9348  acnlem  9970  fin23lem29  10263  itunitc  10343  hsmexlem5  10352  wunfv  10655  wunex2  10661  strfvss  17157  prdsvallem  17417  prdsval  17418  prdsbas  17420  prdsplusg  17421  prdsmulr  17422  prdsvsca  17423  prdshom  17430  mreunirn  17563  mrcfval  17574  mrcssv  17580  mrisval  17596  sscpwex  17782  wunfunc  17868  catcxpccl  18173  comppfsc  23497  filunirn  23847  elflim  23936  flffval  23954  fclsval  23973  isfcls  23974  fcfval  23998  tsmsxplem1  24118  xmetunirn  24302  mopnval  24403  tmsval  24446  cfilfval  25231  caufval  25242  issgon  34267  elrnsiga  34270  volmeas  34375  omssubadd  34444  neibastop2lem  36542  ctbssinf  37722  ismtyval  38121  dicval  41622  prjcrv0  43066  ismrc  43133  nacsfix  43144  hbt  43558