Theorem fnunirn 5859

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 5390	. . 3 |- ( F Fn I -> Fun F )
2		elunirn 5858	. . 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 5392	. . 3 |- ( F Fn I -> dom F = I )
5	4	rexeqdv 2712	. 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 2178   E.wrex 2487   U.cuni 3864   dom cdm 4693   ran crn 4694   Fun wfun 5284   Fn wfn 5285   ` cfv 5290

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298

This theorem is referenced by:  xmetunirn  14945