Theorem fniunfv 5766

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 5534	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
2	1	funfni 5318	. . . 4 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
3	2	ralrimiva 2550	. . 3 |- ( F Fn A -> A. x e. A ( F ` x ) e. _V )
4		dfiun2g 3920	. . 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 5565	. . 3 |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } )
7	6	unieqd 3822	. 2 |- ( F Fn A -> U. ran F = U. { y | E. x e. A y = ( F ` x ) } )
8	5, 7	eqtr4d 2213	1 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ran F )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   _Vcvv 2739   U.cuni 3811   U_ciun 3888   ran crn 4629   Fn wfn 5213   ` cfv 5218

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226

This theorem is referenced by:  funiunfvdm  5767  ennnfonelemfun  12421  ennnfonelemf1  12422