Theorem funiunfvdm 5834

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 5833. (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 5833	. 2 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ran F )
2		imadmrn 5033	. . . 4 |- ( F " dom F ) = ran F
3		fndm 5374	. . . . 5 |- ( F Fn A -> dom F = A )
4	3	imaeq2d 5023	. . . 4 |- ( F Fn A -> ( F " dom F ) = ( F " A ) )
5	2, 4	eqtr3id 2252	. . 3 |- ( F Fn A -> ran F = ( F " A ) )
6	5	unieqd 3861	. 2 |- ( F Fn A -> U. ran F = U. ( F " A ) )
7	1, 6	eqtrd 2238	1 |- ( F Fn A -> U_ x e. A ( F ` x ) = U. ( F " A ) )

Syntax hints:   -> wi 4   = wceq 1373   U.cuni 3850   U_ciun 3927   dom cdm 4676   ran crn 4677   "cima 4679   Fn wfn 5267   ` cfv 5272

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280

This theorem is referenced by:  funiunfvdmf  5835  eluniimadm  5836