Theorem fniunfv 5898

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		funfvex 5652	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
2	1	funfni 5429	. . . 4 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
3	2	ralrimiva 2603	. . 3 |- ( F Fn A -> A. x e. A ( F ` x ) e. _V )
4		dfiun2g 4000	. . 3 |- ( A. x e. A ( F ` x ) e. _V -> U_ x e. A ( F ` x ) = U. { y | E. x e. A y = ( F ` x ) } )
5	3, 4	syl 14	. 2 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. { y | E. x e. A y = ( F ` x ) } )
6		fnrnfv 5688	. . 3 |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } )
7	6	unieqd 3902	. 2 |- ( F Fn A -> U. ran F = U. { y | E. x e. A y = ( F ` x ) } )
8	5, 7	eqtr4d 2265	1 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ran F )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   _Vcvv 2800   U.cuni 3891   U_ciun 3968   ran crn 4724   Fn wfn 5319   ` cfv 5324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332

This theorem is referenced by:  funiunfvdm  5899  ennnfonelemfun  13028  ennnfonelemf1  13029