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Theorem gruiun 9175
Description: If  B
( x ) is a family of elements of  U and the index set  A is an element of  U, then the indexed union  U_ x  e.  A B is also an element of  U, where  U is a Grothendieck universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
gruiun  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  A. x  e.  A  B  e.  U )  ->  U_ x  e.  A  B  e.  U )
Distinct variable groups:    x, U    x, A
Allowed substitution hint:    B( x)

Proof of Theorem gruiun
StepHypRef Expression
1 eqid 2441 . . . . . . 7  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
21fnmpt 5693 . . . . . 6  |-  ( A. x  e.  A  B  e.  U  ->  ( x  e.  A  |->  B )  Fn  A )
31rnmptss 6041 . . . . . 6  |-  ( A. x  e.  A  B  e.  U  ->  ran  (
x  e.  A  |->  B )  C_  U )
4 df-f 5578 . . . . . 6  |-  ( ( x  e.  A  |->  B ) : A --> U  <->  ( (
x  e.  A  |->  B )  Fn  A  /\  ran  ( x  e.  A  |->  B )  C_  U
) )
52, 3, 4sylanbrc 664 . . . . 5  |-  ( A. x  e.  A  B  e.  U  ->  ( x  e.  A  |->  B ) : A --> U )
6 gruurn 9174 . . . . . 6  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  (
x  e.  A  |->  B ) : A --> U )  ->  U. ran  ( x  e.  A  |->  B )  e.  U )
763expia 1197 . . . . 5  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( x  e.  A  |->  B ) : A --> U  ->  U. ran  ( x  e.  A  |->  B )  e.  U ) )
85, 7syl5com 30 . . . 4  |-  ( A. x  e.  A  B  e.  U  ->  ( ( U  e.  Univ  /\  A  e.  U )  ->  U. ran  ( x  e.  A  |->  B )  e.  U
) )
9 dfiun3g 5241 . . . . 5  |-  ( A. x  e.  A  B  e.  U  ->  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B ) )
109eleq1d 2510 . . . 4  |-  ( A. x  e.  A  B  e.  U  ->  ( U_ x  e.  A  B  e.  U  <->  U. ran  ( x  e.  A  |->  B )  e.  U ) )
118, 10sylibrd 234 . . 3  |-  ( A. x  e.  A  B  e.  U  ->  ( ( U  e.  Univ  /\  A  e.  U )  ->  U_ x  e.  A  B  e.  U ) )
1211com12 31 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( A. x  e.  A  B  e.  U  ->  U_ x  e.  A  B  e.  U ) )
13123impia 1192 1  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  A. x  e.  A  B  e.  U )  ->  U_ x  e.  A  B  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    e. wcel 1802   A.wral 2791    C_ wss 3458   U.cuni 4230   U_ciun 4311    |-> cmpt 4491   ran crn 4986    Fn wfn 5569   -->wf 5570   Univcgru 9166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7420  df-gru 9167
This theorem is referenced by:  gruuni  9176  gruun  9182  gruixp  9185  grur1a  9195
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