Theorem fvssunirn 6127

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	⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹

Proof of Theorem fvssunirn

Step	Hyp	Ref	Expression
1		fvrn0 6126	. . 3 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})
2		elssuni 4403	. . 3 ⊢ ((𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) → (𝐹‘𝑋) ⊆ ∪ (ran 𝐹 ∪ {∅}))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐹‘𝑋) ⊆ ∪ (ran 𝐹 ∪ {∅})
4		uniun 4392	. . 3 ⊢ ∪ (ran 𝐹 ∪ {∅}) = (∪ ran 𝐹 ∪ ∪ {∅})
5		0ex 4718	. . . . 5 ⊢ ∅ ∈ V
6	5	unisn 4387	. . . 4 ⊢ ∪ {∅} = ∅
7	6	uneq2i 3726	. . 3 ⊢ (∪ ran 𝐹 ∪ ∪ {∅}) = (∪ ran 𝐹 ∪ ∅)
8		un0 3919	. . 3 ⊢ (∪ ran 𝐹 ∪ ∅) = ∪ ran 𝐹
9	4, 7, 8	3eqtri 2636	. 2 ⊢ ∪ (ran 𝐹 ∪ {∅}) = ∪ ran 𝐹
10	3, 9	sseqtri 3600	1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹

Syntax hints:   ∈ wcel 1977   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  {csn 4125  ∪ cuni 4372  ran crn 5039  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-cnv 5046  df-dm 5048  df-rn 5049  df-iota 5768  df-fv 5812

This theorem is referenced by:  ovssunirn  6579  marypha2lem1  8224  acnlem  8754  fin23lem29  9046  itunitc  9126  hsmexlem5  9135  wunfv  9433  wunex2  9439  strfvss  15713  prdsval  15938  prdsbas  15940  prdsplusg  15941  prdsmulr  15942  prdsvsca  15943  prdshom  15950  mreunirn  16084  mrcfval  16091  mrcssv  16097  mrisval  16113  sscpwex  16298  wunfunc  16382  catcxpccl  16670  comppfsc  21145  filunirn  21496  elflim  21585  flffval  21603  fclsval  21622  isfcls  21623  fcfval  21647  tsmsxplem1  21766  xmetunirn  21952  mopnval  22053  tmsval  22096  cfilfval  22870  caufval  22881  issgon  29513  elrnsiga  29516  volmeas  29621  omssubadd  29689  neibastop2lem  31525  ismtyval  32769  dicval  35483  ismrc  36282  nacsfix  36293  hbt  36719