Theorem fniunfv 6505

Description: The indexed union of a function's values is the union of its range. Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM, 27-Sep-2004.)

Assertion

Ref	Expression
fniunfv	|- ( F Fn A -> U_ x e. A ( F ` x ) = U. ran F )

Distinct variable groups:   x, A   x, F

Proof of Theorem fniunfv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnrnfv 6242	. . 3 |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } )
2	1	unieqd 4446	. 2 |- ( F Fn A -> U. ran F = U. { y | E. x e. A y = ( F ` x ) } )
3		fvex 6201	. . 3 |- ( F ` x ) e. _V
4	3	dfiun2 4554	. 2 |- U_ x e. A ( F ` x ) = U. { y | E. x e. A y = ( F ` x ) }
5	2, 4	syl6reqr 2675	1 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ran F )

Syntax hints:   -> wi 4   = wceq 1483   {cab 2608   E.wrex 2913   U.cuni 4436   U_ciun 4520   ran crn 5115   Fn wfn 5883   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  funiunfv  6506  dffi3  8337  jech9.3  8677  hsmexlem5  9252  wuncval2  9569  dprdspan  18426  tgcmp  21204  txcmplem1  21444  txcmplem2  21445  xkococnlem  21462  alexsubALT  21855  bcth3  23128  ovolfioo  23236  ovolficc  23237  voliunlem2  23319  voliunlem3  23320  volsup  23324  uniiccdif  23346  uniioovol  23347  uniiccvol  23348  uniioombllem2  23351  uniioombllem4  23354  volsup2  23373  itg1climres  23481  itg2monolem1  23517  itg2gt0  23527  sigapildsys  30225  omssubadd  30362  carsgclctunlem3  30382  dftrpred2  31719  volsupnfl  33454  hbt  37700  ovolval4lem1  40863  ovolval5lem3  40868  ovnovollem1  40870  ovnovollem2  40871