Theorem fniunfv 5741

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 5513	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
2	1	funfni 5298	. . . 4 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
3	2	ralrimiva 2543	. . 3 |- ( F Fn A -> A. x e. A ( F ` x ) e. _V )
4		dfiun2g 3905	. . 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 5543	. . 3 |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } )
7	6	unieqd 3807	. 2 |- ( F Fn A -> U. ran F = U. { y | E. x e. A y = ( F ` x ) } )
8	5, 7	eqtr4d 2206	1 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ran F )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   {cab 2156   A.wral 2448   E.wrex 2449   _Vcvv 2730   U.cuni 3796   U_ciun 3873   ran crn 4612   Fn wfn 5193   ` cfv 5198

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by:  funiunfvdm  5742  ennnfonelemfun  12372  ennnfonelemf1  12373