Theorem fvssunirn 6914

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 6913	. 2 ⊢ (𝑥 ∈ (𝐹‘𝑋) → 𝑥 ∈ ∪ ran 𝐹)
2	1	ssriv 3967	1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹

Syntax hints:   ⊆ wss 3931  ∪ cuni 4888  ran crn 5660  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-cnv 5667  df-dm 5669  df-rn 5670  df-iota 6489  df-fv 6544

This theorem is referenced by:  ovssunirn  7446  marypha2lem1  9452  acnlem  10067  fin23lem29  10360  itunitc  10440  hsmexlem5  10449  wunfv  10751  wunex2  10757  strfvss  17211  prdsvallem  17473  prdsval  17474  prdsbas  17476  prdsplusg  17477  prdsmulr  17478  prdsvsca  17479  prdshom  17486  mreunirn  17618  mrcfval  17625  mrcssv  17631  mrisval  17647  sscpwex  17833  wunfunc  17919  catcxpccl  18224  comppfsc  23475  filunirn  23825  elflim  23914  flffval  23932  fclsval  23951  isfcls  23952  fcfval  23976  tsmsxplem1  24096  xmetunirn  24281  mopnval  24382  tmsval  24425  cfilfval  25221  caufval  25232  issgon  34159  elrnsiga  34162  volmeas  34267  omssubadd  34337  neibastop2lem  36383  ctbssinf  37429  ismtyval  37829  dicval  41200  prjcrv0  42623  ismrc  42691  nacsfix  42702  hbt  43121