Theorem fvssunirn 6940

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 6939	. 2 ⊢ (𝑥 ∈ (𝐹‘𝑋) → 𝑥 ∈ ∪ ran 𝐹)
2	1	ssriv 3999	1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹

Syntax hints:   ⊆ wss 3963  ∪ cuni 4912  ran crn 5690  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700  df-iota 6516  df-fv 6571

This theorem is referenced by:  ovssunirn  7467  marypha2lem1  9473  acnlem  10086  fin23lem29  10379  itunitc  10459  hsmexlem5  10468  wunfv  10770  wunex2  10776  strfvss  17221  prdsvallem  17501  prdsval  17502  prdsbas  17504  prdsplusg  17505  prdsmulr  17506  prdsvsca  17507  prdshom  17514  mreunirn  17646  mrcfval  17653  mrcssv  17659  mrisval  17675  sscpwex  17863  wunfunc  17952  wunfuncOLD  17953  catcxpccl  18263  catcxpcclOLD  18264  comppfsc  23556  filunirn  23906  elflim  23995  flffval  24013  fclsval  24032  isfcls  24033  fcfval  24057  tsmsxplem1  24177  xmetunirn  24363  mopnval  24464  tmsval  24509  cfilfval  25312  caufval  25323  issgon  34104  elrnsiga  34107  volmeas  34212  omssubadd  34282  neibastop2lem  36343  ctbssinf  37389  ismtyval  37787  dicval  41159  prjcrv0  42620  ismrc  42689  nacsfix  42700  hbt  43119