Theorem fvssunirn 6867

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 6866	. 2 ⊢ (𝑥 ∈ (𝐹‘𝑋) → 𝑥 ∈ ∪ ran 𝐹)
2	1	ssriv 3926	1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹

Syntax hints:   ⊆ wss 3890  ∪ cuni 4851  ran crn 5627  ‘cfv 6494

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-cnv 5634  df-dm 5636  df-rn 5637  df-iota 6450  df-fv 6502

This theorem is referenced by:  ovssunirn  7398  marypha2lem1  9343  acnlem  9965  fin23lem29  10258  itunitc  10338  hsmexlem5  10347  wunfv  10650  wunex2  10656  strfvss  17152  prdsvallem  17412  prdsval  17413  prdsbas  17415  prdsplusg  17416  prdsmulr  17417  prdsvsca  17418  prdshom  17425  mreunirn  17558  mrcfval  17569  mrcssv  17575  mrisval  17591  sscpwex  17777  wunfunc  17863  catcxpccl  18168  comppfsc  23511  filunirn  23861  elflim  23950  flffval  23968  fclsval  23987  isfcls  23988  fcfval  24012  tsmsxplem1  24132  xmetunirn  24316  mopnval  24417  tmsval  24460  cfilfval  25245  caufval  25256  issgon  34287  elrnsiga  34290  volmeas  34395  omssubadd  34464  neibastop2lem  36562  ctbssinf  37742  ismtyval  38141  dicval  41642  prjcrv0  43086  ismrc  43153  nacsfix  43164  hbt  43582