Theorem fniunfv 5703

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 5478	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
2	1	funfni 5263	. . . 4 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
3	2	ralrimiva 2527	. . 3 |- ( F Fn A -> A. x e. A ( F ` x ) e. _V )
4		dfiun2g 3877	. . 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 5508	. . 3 |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } )
7	6	unieqd 3779	. 2 |- ( F Fn A -> U. ran F = U. { y | E. x e. A y = ( F ` x ) } )
8	5, 7	eqtr4d 2190	1 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ran F )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 2125   {cab 2140   A.wral 2432   E.wrex 2433   _Vcvv 2709   U.cuni 3768   U_ciun 3845   ran crn 4580   Fn wfn 5158   ` cfv 5163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-sbc 2934  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-iota 5128  df-fun 5165  df-fn 5166  df-fv 5171

This theorem is referenced by:  funiunfvdm  5704  ennnfonelemfun  12097  ennnfonelemf1  12098