Theorem fvssunirn 6785

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 6784	. . 3 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})
2		elssuni 4868	. . 3 ⊢ ((𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) → (𝐹‘𝑋) ⊆ ∪ (ran 𝐹 ∪ {∅}))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐹‘𝑋) ⊆ ∪ (ran 𝐹 ∪ {∅})
4		uniun 4861	. . 3 ⊢ ∪ (ran 𝐹 ∪ {∅}) = (∪ ran 𝐹 ∪ ∪ {∅})
5		0ex 5226	. . . . 5 ⊢ ∅ ∈ V
6	5	unisn 4858	. . . 4 ⊢ ∪ {∅} = ∅
7	6	uneq2i 4090	. . 3 ⊢ (∪ ran 𝐹 ∪ ∪ {∅}) = (∪ ran 𝐹 ∪ ∅)
8		un0 4321	. . 3 ⊢ (∪ ran 𝐹 ∪ ∅) = ∪ ran 𝐹
9	4, 7, 8	3eqtri 2770	. 2 ⊢ ∪ (ran 𝐹 ∪ {∅}) = ∪ ran 𝐹
10	3, 9	sseqtri 3953	1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹

Syntax hints:   ∈ wcel 2108   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  {csn 4558  ∪ cuni 4836  ran crn 5581  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-cnv 5588  df-dm 5590  df-rn 5591  df-iota 6376  df-fv 6426

This theorem is referenced by:  ovssunirn  7291  marypha2lem1  9124  acnlem  9735  fin23lem29  10028  itunitc  10108  hsmexlem5  10117  wunfv  10419  wunex2  10425  strfvss  16816  prdsvallem  17082  prdsval  17083  prdsbas  17085  prdsplusg  17086  prdsmulr  17087  prdsvsca  17088  prdshom  17095  mreunirn  17227  mrcfval  17234  mrcssv  17240  mrisval  17256  sscpwex  17444  wunfunc  17530  wunfuncOLD  17531  catcxpccl  17840  catcxpcclOLD  17841  comppfsc  22591  filunirn  22941  elflim  23030  flffval  23048  fclsval  23067  isfcls  23068  fcfval  23092  tsmsxplem1  23212  xmetunirn  23398  mopnval  23499  tmsval  23542  cfilfval  24333  caufval  24344  issgon  31991  elrnsiga  31994  volmeas  32099  omssubadd  32167  neibastop2lem  34476  ctbssinf  35504  ismtyval  35885  dicval  39117  ismrc  40439  nacsfix  40450  hbt  40871