Theorem fvssunirn 6894

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 6893	. 2 ⊢ (𝑥 ∈ (𝐹‘𝑋) → 𝑥 ∈ ∪ ran 𝐹)
2	1	ssriv 3953	1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹

Syntax hints:   ⊆ wss 3917  ∪ cuni 4874  ran crn 5642  ‘cfv 6514

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-cnv 5649  df-dm 5651  df-rn 5652  df-iota 6467  df-fv 6522

This theorem is referenced by:  ovssunirn  7426  marypha2lem1  9393  acnlem  10008  fin23lem29  10301  itunitc  10381  hsmexlem5  10390  wunfv  10692  wunex2  10698  strfvss  17164  prdsvallem  17424  prdsval  17425  prdsbas  17427  prdsplusg  17428  prdsmulr  17429  prdsvsca  17430  prdshom  17437  mreunirn  17569  mrcfval  17576  mrcssv  17582  mrisval  17598  sscpwex  17784  wunfunc  17870  catcxpccl  18175  comppfsc  23426  filunirn  23776  elflim  23865  flffval  23883  fclsval  23902  isfcls  23903  fcfval  23927  tsmsxplem1  24047  xmetunirn  24232  mopnval  24333  tmsval  24376  cfilfval  25171  caufval  25182  issgon  34120  elrnsiga  34123  volmeas  34228  omssubadd  34298  neibastop2lem  36355  ctbssinf  37401  ismtyval  37801  dicval  41177  prjcrv0  42628  ismrc  42696  nacsfix  42707  hbt  43126