Theorem fvssunirn 6924

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 6923	. 2 ⊢ (𝑥 ∈ (𝐹‘𝑋) → 𝑥 ∈ ∪ ran 𝐹)
2	1	ssriv 3986	1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹

Syntax hints:   ⊆ wss 3948  ∪ cuni 4908  ran crn 5677  ‘cfv 6543

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-cnv 5684  df-dm 5686  df-rn 5687  df-iota 6495  df-fv 6551

This theorem is referenced by:  ovssunirn  7447  marypha2lem1  9432  acnlem  10045  fin23lem29  10338  itunitc  10418  hsmexlem5  10427  wunfv  10729  wunex2  10735  strfvss  17122  prdsvallem  17402  prdsval  17403  prdsbas  17405  prdsplusg  17406  prdsmulr  17407  prdsvsca  17408  prdshom  17415  mreunirn  17547  mrcfval  17554  mrcssv  17560  mrisval  17576  sscpwex  17764  wunfunc  17851  wunfuncOLD  17852  catcxpccl  18161  catcxpcclOLD  18162  comppfsc  23043  filunirn  23393  elflim  23482  flffval  23500  fclsval  23519  isfcls  23520  fcfval  23544  tsmsxplem1  23664  xmetunirn  23850  mopnval  23951  tmsval  23996  cfilfval  24788  caufval  24799  issgon  33190  elrnsiga  33193  volmeas  33298  omssubadd  33368  neibastop2lem  35331  ctbssinf  36373  ismtyval  36754  dicval  40133  prjcrv0  41457  ismrc  41521  nacsfix  41532  hbt  41954