Theorem fvssunirn 6916

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 6915	. 2 ⊢ (𝑥 ∈ (𝐹‘𝑋) → 𝑥 ∈ ∪ ran 𝐹)
2	1	ssriv 3912	1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹

This proof depends on syntax axioms:   ⊆ wss 3895  ∪ cuni 4866  ran crn 5651  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-cnv 5658  df-dm 5660  df-rn 5661  df-iota 6494  df-fv 6546

This theorem is used by:  ovssunirn  7456  marypha2lem1  9427  acnlem  10127  fin23lem29  10419  itunitc  10499  hsmexlem5  10508  wunfv  10817  wunex2  10823  strfvss  17365  prdsvallem  17625  prdsval  17626  prdsbas  17628  prdsplusg  17629  prdsmulr  17630  prdsvsca  17631  prdshom  17638  mreunirn  17771  mrcfval  17782  mrcssv  17788  mrisval  17804  sscpwex  17990  wunfunc  18076  catcxpccl  18381  comppfsc  23851  filunirn  24201  elflim  24290  flffval  24308  fclsval  24327  isfcls  24328  fcfval  24352  tsmsxplem1  24472  xmetunirn  24656  mopnval  24757  tmsval  24800  cfilfval  25585  caufval  25596  issgon  34755  elrnsiga  34758  volmeas  34864  omssubadd  34932  neibastop2lem  37148  ctbssinf  38330  ismtyval  38735  dicval  42234  prjcrv0  43670  ismrc  43712  nacsfix  43723  hbt  44132