Theorem fvssunirn 6697

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 6696	. . 3 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})
2		elssuni 4825	. . 3 ⊢ ((𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) → (𝐹‘𝑋) ⊆ ∪ (ran 𝐹 ∪ {∅}))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐹‘𝑋) ⊆ ∪ (ran 𝐹 ∪ {∅})
4		uniun 4818	. . 3 ⊢ ∪ (ran 𝐹 ∪ {∅}) = (∪ ran 𝐹 ∪ ∪ {∅})
5		0ex 5172	. . . . 5 ⊢ ∅ ∈ V
6	5	unisn 4815	. . . 4 ⊢ ∪ {∅} = ∅
7	6	uneq2i 4048	. . 3 ⊢ (∪ ran 𝐹 ∪ ∪ {∅}) = (∪ ran 𝐹 ∪ ∅)
8		un0 4276	. . 3 ⊢ (∪ ran 𝐹 ∪ ∅) = ∪ ran 𝐹
9	4, 7, 8	3eqtri 2765	. 2 ⊢ ∪ (ran 𝐹 ∪ {∅}) = ∪ ran 𝐹
10	3, 9	sseqtri 3911	1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹

Syntax hints:   ∈ wcel 2113   ∪ cun 3839   ⊆ wss 3841  ∅c0 4209  {csn 4513  ∪ cuni 4793  ran crn 5520  ‘cfv 6333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-ral 3058  df-rex 3059  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-cnv 5527  df-dm 5529  df-rn 5530  df-iota 6291  df-fv 6341

This theorem is referenced by:  ovssunirn  7200  marypha2lem1  8965  acnlem  9541  fin23lem29  9834  itunitc  9914  hsmexlem5  9923  wunfv  10225  wunex2  10231  strfvss  16602  prdsval  16824  prdsbas  16826  prdsplusg  16827  prdsmulr  16828  prdsvsca  16829  prdshom  16836  mreunirn  16968  mrcfval  16975  mrcssv  16981  mrisval  16997  sscpwex  17183  wunfunc  17267  catcxpccl  17566  comppfsc  22276  filunirn  22626  elflim  22715  flffval  22733  fclsval  22752  isfcls  22753  fcfval  22777  tsmsxplem1  22897  xmetunirn  23083  mopnval  23184  tmsval  23227  cfilfval  24009  caufval  24020  issgon  31653  elrnsiga  31656  volmeas  31761  omssubadd  31829  neibastop2lem  34179  ctbssinf  35189  ismtyval  35570  dicval  38802  ismrc  40079  nacsfix  40090  hbt  40511