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Theorem fvssunirn 6174
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.)
Assertion
Ref Expression
fvssunirn (𝐹𝑋) ⊆ ran 𝐹

Proof of Theorem fvssunirn
StepHypRef Expression
1 fvrn0 6173 . . 3 (𝐹𝑋) ∈ (ran 𝐹 ∪ {∅})
2 elssuni 4433 . . 3 ((𝐹𝑋) ∈ (ran 𝐹 ∪ {∅}) → (𝐹𝑋) ⊆ (ran 𝐹 ∪ {∅}))
31, 2ax-mp 5 . 2 (𝐹𝑋) ⊆ (ran 𝐹 ∪ {∅})
4 uniun 4422 . . 3 (ran 𝐹 ∪ {∅}) = ( ran 𝐹 {∅})
5 0ex 4750 . . . . 5 ∅ ∈ V
65unisn 4417 . . . 4 {∅} = ∅
76uneq2i 3742 . . 3 ( ran 𝐹 {∅}) = ( ran 𝐹 ∪ ∅)
8 un0 3939 . . 3 ( ran 𝐹 ∪ ∅) = ran 𝐹
94, 7, 83eqtri 2647 . 2 (ran 𝐹 ∪ {∅}) = ran 𝐹
103, 9sseqtri 3616 1 (𝐹𝑋) ⊆ ran 𝐹
Colors of variables: wff setvar class
Syntax hints:  wcel 1987  cun 3553  wss 3555  c0 3891  {csn 4148   cuni 4402  ran crn 5075  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-cnv 5082  df-dm 5084  df-rn 5085  df-iota 5810  df-fv 5855
This theorem is referenced by:  ovssunirn  6634  marypha2lem1  8285  acnlem  8815  fin23lem29  9107  itunitc  9187  hsmexlem5  9196  wunfv  9498  wunex2  9504  strfvss  15802  prdsval  16036  prdsbas  16038  prdsplusg  16039  prdsmulr  16040  prdsvsca  16041  prdshom  16048  mreunirn  16182  mrcfval  16189  mrcssv  16195  mrisval  16211  sscpwex  16396  wunfunc  16480  catcxpccl  16768  comppfsc  21245  filunirn  21596  elflim  21685  flffval  21703  fclsval  21722  isfcls  21723  fcfval  21747  tsmsxplem1  21866  xmetunirn  22052  mopnval  22153  tmsval  22196  cfilfval  22970  caufval  22981  issgon  29967  elrnsiga  29970  volmeas  30075  omssubadd  30143  neibastop2lem  31997  ismtyval  33231  dicval  35945  ismrc  36744  nacsfix  36755  hbt  37181
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