Theorem fvssunirn 6674

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 6673	. . 3 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})
2		elssuni 4830	. . 3 ⊢ ((𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) → (𝐹‘𝑋) ⊆ ∪ (ran 𝐹 ∪ {∅}))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐹‘𝑋) ⊆ ∪ (ran 𝐹 ∪ {∅})
4		uniun 4823	. . 3 ⊢ ∪ (ran 𝐹 ∪ {∅}) = (∪ ran 𝐹 ∪ ∪ {∅})
5		0ex 5175	. . . . 5 ⊢ ∅ ∈ V
6	5	unisn 4820	. . . 4 ⊢ ∪ {∅} = ∅
7	6	uneq2i 4087	. . 3 ⊢ (∪ ran 𝐹 ∪ ∪ {∅}) = (∪ ran 𝐹 ∪ ∅)
8		un0 4298	. . 3 ⊢ (∪ ran 𝐹 ∪ ∅) = ∪ ran 𝐹
9	4, 7, 8	3eqtri 2825	. 2 ⊢ ∪ (ran 𝐹 ∪ {∅}) = ∪ ran 𝐹
10	3, 9	sseqtri 3951	1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹

Syntax hints:   ∈ wcel 2111   ∪ cun 3879   ⊆ wss 3881  ∅c0 4243  {csn 4525  ∪ cuni 4800  ran crn 5520  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-cnv 5527  df-dm 5529  df-rn 5530  df-iota 6283  df-fv 6332

This theorem is referenced by:  ovssunirn  7171  marypha2lem1  8883  acnlem  9459  fin23lem29  9752  itunitc  9832  hsmexlem5  9841  wunfv  10143  wunex2  10149  strfvss  16498  prdsval  16720  prdsbas  16722  prdsplusg  16723  prdsmulr  16724  prdsvsca  16725  prdshom  16732  mreunirn  16864  mrcfval  16871  mrcssv  16877  mrisval  16893  sscpwex  17077  wunfunc  17161  catcxpccl  17449  comppfsc  22137  filunirn  22487  elflim  22576  flffval  22594  fclsval  22613  isfcls  22614  fcfval  22638  tsmsxplem1  22758  xmetunirn  22944  mopnval  23045  tmsval  23088  cfilfval  23868  caufval  23879  issgon  31492  elrnsiga  31495  volmeas  31600  omssubadd  31668  neibastop2lem  33821  ctbssinf  34823  ismtyval  35238  dicval  38472  ismrc  39642  nacsfix  39653  hbt  40074