Theorem fnunirn 5661

Description: Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Assertion

Ref	Expression
fnunirn	|- ( F Fn I -> ( A e. U. ran F <-> E. x e. I A e. ( F ` x ) ) )

Distinct variable groups:   x, A   x, I   x, F

Proof of Theorem fnunirn

Step	Hyp	Ref	Expression
1		fnfun 5215	. . 3 |- ( F Fn I -> Fun F )
2		elunirn 5660	. . 3 |- ( Fun F -> ( A e. U. ran F <-> E. x e. dom F A e. ( F ` x ) ) )
3	1, 2	syl 14	. 2 |- ( F Fn I -> ( A e. U. ran F <-> E. x e. dom F A e. ( F ` x ) ) )
4		fndm 5217	. . 3 |- ( F Fn I -> dom F = I )
5	4	rexeqdv 2631	. 2 |- ( F Fn I -> ( E. x e. dom F A e. ( F ` x ) <-> E. x e. I A e. ( F ` x ) ) )
6	3, 5	bitrd 187	1 |- ( F Fn I -> ( A e. U. ran F <-> E. x e. I A e. ( F ` x ) ) )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 1480   E.wrex 2415   U.cuni 3731   dom cdm 4534   ran crn 4535   Fun wfun 5112   Fn wfn 5113   ` cfv 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126

This theorem is referenced by:  xmetunirn  12516