Theorem funiunfvdm 5807

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 5806. (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 5806	. 2 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ran F )
2		imadmrn 5016	. . . 4 |- ( F " dom F ) = ran F
3		fndm 5354	. . . . 5 |- ( F Fn A -> dom F = A )
4	3	imaeq2d 5006	. . . 4 |- ( F Fn A -> ( F " dom F ) = ( F " A ) )
5	2, 4	eqtr3id 2240	. . 3 |- ( F Fn A -> ran F = ( F " A ) )
6	5	unieqd 3847	. 2 |- ( F Fn A -> U. ran F = U. ( F " A ) )
7	1, 6	eqtrd 2226	1 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ( F " A ) )

Syntax hints:   -> wi 4   = wceq 1364   U.cuni 3836   U_ciun 3913   dom cdm 4660   ran crn 4661   "cima 4663   Fn wfn 5250   ` cfv 5255

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263

This theorem is referenced by:  funiunfvdmf  5808  eluniimadm  5809