Theorem fnunirn 5940

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 5453	. . 3 |- ( F Fn I -> Fun F )
2		elunirn 5939	. . 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 5455	. . 3 |- ( F Fn I -> dom F = I )
5	4	rexeqdv 2748	. 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 188	1 |- ( F Fn I -> ( A e. U. ran F <-> E. x e. I A e. ( F ` x ) ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2203   E.wrex 2521   U.cuni 3914   dom cdm 4749   ran crn 4750   Fun wfun 5346   Fn wfn 5347   ` cfv 5352

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360

This theorem is referenced by:  xmetunirn  15223