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Theorem gruiun 8958
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
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqid 2437 . . . . . . 7  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
21fnmpt 5530 . . . . . 6  |-  ( A. x  e.  A  B  e.  U  ->  ( x  e.  A  |->  B )  Fn  A )
31rnmpt 5077 . . . . . . 7  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
4 r19.29 2851 . . . . . . . . . 10  |-  ( ( A. x  e.  A  B  e.  U  /\  E. x  e.  A  y  =  B )  ->  E. x  e.  A  ( B  e.  U  /\  y  =  B
) )
5 eleq1 2497 . . . . . . . . . . . 12  |-  ( y  =  B  ->  (
y  e.  U  <->  B  e.  U ) )
65biimparc 487 . . . . . . . . . . 11  |-  ( ( B  e.  U  /\  y  =  B )  ->  y  e.  U )
76rexlimivw 2831 . . . . . . . . . 10  |-  ( E. x  e.  A  ( B  e.  U  /\  y  =  B )  ->  y  e.  U )
84, 7syl 16 . . . . . . . . 9  |-  ( ( A. x  e.  A  B  e.  U  /\  E. x  e.  A  y  =  B )  -> 
y  e.  U )
98ex 434 . . . . . . . 8  |-  ( A. x  e.  A  B  e.  U  ->  ( E. x  e.  A  y  =  B  ->  y  e.  U ) )
109abssdv 3419 . . . . . . 7  |-  ( A. x  e.  A  B  e.  U  ->  { y  |  E. x  e.  A  y  =  B }  C_  U )
113, 10syl5eqss 3393 . . . . . 6  |-  ( A. x  e.  A  B  e.  U  ->  ran  (
x  e.  A  |->  B )  C_  U )
12 df-f 5415 . . . . . 6  |-  ( ( x  e.  A  |->  B ) : A --> U  <->  ( (
x  e.  A  |->  B )  Fn  A  /\  ran  ( x  e.  A  |->  B )  C_  U
) )
132, 11, 12sylanbrc 664 . . . . 5  |-  ( A. x  e.  A  B  e.  U  ->  ( x  e.  A  |->  B ) : A --> U )
14 gruurn 8957 . . . . . 6  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  (
x  e.  A  |->  B ) : A --> U )  ->  U. ran  ( x  e.  A  |->  B )  e.  U )
15143expia 1189 . . . . 5  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( x  e.  A  |->  B ) : A --> U  ->  U. ran  ( x  e.  A  |->  B )  e.  U ) )
1613, 15syl5com 30 . . . 4  |-  ( A. x  e.  A  B  e.  U  ->  ( ( U  e.  Univ  /\  A  e.  U )  ->  U. ran  ( x  e.  A  |->  B )  e.  U
) )
17 dfiun3g 5084 . . . . 5  |-  ( A. x  e.  A  B  e.  U  ->  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B ) )
1817eleq1d 2503 . . . 4  |-  ( A. x  e.  A  B  e.  U  ->  ( U_ x  e.  A  B  e.  U  <->  U. ran  ( x  e.  A  |->  B )  e.  U ) )
1916, 18sylibrd 234 . . 3  |-  ( A. x  e.  A  B  e.  U  ->  ( ( U  e.  Univ  /\  A  e.  U )  ->  U_ x  e.  A  B  e.  U ) )
2019com12 31 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( A. x  e.  A  B  e.  U  ->  U_ x  e.  A  B  e.  U ) )
21203impia 1184 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 965    = wceq 1369    e. wcel 1756   {cab 2423   A.wral 2709   E.wrex 2710    C_ wss 3321   U.cuni 4084   U_ciun 4164    e. cmpt 4343   ran crn 4833    Fn wfn 5406   -->wf 5407   Univcgru 8949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2714  df-rex 2715  df-rab 2718  df-v 2968  df-sbc 3180  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-op 3877  df-uni 4085  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-id 4628  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-fv 5419  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-map 7208  df-gru 8950
This theorem is referenced by:  gruuni  8959  gruun  8965  gruixp  8968  grur1a  8978
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