Theorem funiunfvdm 5596

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 5595. (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 5595	. 2 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ran F )
2		imadmrn 4827	. . . 4 |- ( F " dom F ) = ran F
3		fndm 5158	. . . . 5 |- ( F Fn A -> dom F = A )
4	3	imaeq2d 4817	. . . 4 |- ( F Fn A -> ( F " dom F ) = ( F " A ) )
5	2, 4	syl5eqr 2146	. . 3 |- ( F Fn A -> ran F = ( F " A ) )
6	5	unieqd 3694	. 2 |- ( F Fn A -> U. ran F = U. ( F " A ) )
7	1, 6	eqtrd 2132	1 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ( F " A ) )

Syntax hints:   -> wi 4   = wceq 1299   U.cuni 3683   U_ciun 3760   dom cdm 4477   ran crn 4478   "cima 4480   Fn wfn 5054   ` cfv 5059

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-fv 5067

This theorem is referenced by:  funiunfvdmf  5597  eluniimadm  5598