Theorem fvssunirn 6953

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 6952	. 2 ⊢ (𝑥 ∈ (𝐹‘𝑋) → 𝑥 ∈ ∪ ran 𝐹)
2	1	ssriv 4012	1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹

Syntax hints:   ⊆ wss 3976  ∪ cuni 4931  ran crn 5701  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-cnv 5708  df-dm 5710  df-rn 5711  df-iota 6525  df-fv 6581

This theorem is referenced by:  ovssunirn  7484  marypha2lem1  9504  acnlem  10117  fin23lem29  10410  itunitc  10490  hsmexlem5  10499  wunfv  10801  wunex2  10807  strfvss  17234  prdsvallem  17514  prdsval  17515  prdsbas  17517  prdsplusg  17518  prdsmulr  17519  prdsvsca  17520  prdshom  17527  mreunirn  17659  mrcfval  17666  mrcssv  17672  mrisval  17688  sscpwex  17876  wunfunc  17965  wunfuncOLD  17966  catcxpccl  18276  catcxpcclOLD  18277  comppfsc  23561  filunirn  23911  elflim  24000  flffval  24018  fclsval  24037  isfcls  24038  fcfval  24062  tsmsxplem1  24182  xmetunirn  24368  mopnval  24469  tmsval  24514  cfilfval  25317  caufval  25328  issgon  34087  elrnsiga  34090  volmeas  34195  omssubadd  34265  neibastop2lem  36326  ctbssinf  37372  ismtyval  37760  dicval  41133  prjcrv0  42588  ismrc  42657  nacsfix  42668  hbt  43087