Theorem funiunfvdm 5962

Description: The indexed union of a function's values is the union of its image under the index class. This theorem is a slight variation of fniunfv 5961. (Contributed by Jim Kingdon, 10-Jan-2019.)

Assertion

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

Distinct variable groups:   x, A   x, F

Proof of Theorem funiunfvdm

Step	Hyp	Ref	Expression
1		fniunfv 5961	. 2 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ran F )
2		imadmrn 5134	. . . 4 |- ( F " dom F ) = ran F
3		fndm 5478	. . . . 5 |- ( F Fn A -> dom F = A )
4	3	imaeq2d 5124	. . . 4 |- ( F Fn A -> ( F " dom F ) = ( F " A ) )
5	2, 4	eqtr3id 2285	. . 3 |- ( F Fn A -> ran F = ( F " A ) )
6	5	unieqd 3944	. 2 |- ( F Fn A -> U. ran F = U. ( F " A ) )
7	1, 6	eqtrd 2271	1 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ( F " A ) )

Syntax hints:   -> wi 4   = wceq 1402   U.cuni 3933   U_ciun 4010   dom cdm 4772   ran crn 4773   "cima 4775   Fn wfn 5370   ` cfv 5375

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383

This theorem is referenced by:  funiunfvdmf  5963  eluniimadm  5964