Theorem funiunfvdm 5810

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 5809. (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 5809	. 2 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ran F )
2		imadmrn 5019	. . . 4 |- ( F " dom F ) = ran F
3		fndm 5357	. . . . 5 |- ( F Fn A -> dom F = A )
4	3	imaeq2d 5009	. . . 4 |- ( F Fn A -> ( F " dom F ) = ( F " A ) )
5	2, 4	eqtr3id 2243	. . 3 |- ( F Fn A -> ran F = ( F " A ) )
6	5	unieqd 3850	. 2 |- ( F Fn A -> U. ran F = U. ( F " A ) )
7	1, 6	eqtrd 2229	1 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ( F " A ) )

Syntax hints:   -> wi 4   = wceq 1364   U.cuni 3839   U_ciun 3916   dom cdm 4663   ran crn 4664   "cima 4666   Fn wfn 5253   ` cfv 5258

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266

This theorem is referenced by:  funiunfvdmf  5811  eluniimadm  5812