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Theorem gruiun 9166
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 2460 . . . . . . 7  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
21fnmpt 5698 . . . . . 6  |-  ( A. x  e.  A  B  e.  U  ->  ( x  e.  A  |->  B )  Fn  A )
31rnmpt 5239 . . . . . . 7  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
4 r19.29 2990 . . . . . . . . . 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 2532 . . . . . . . . . . . 12  |-  ( y  =  B  ->  (
y  e.  U  <->  B  e.  U ) )
65biimparc 487 . . . . . . . . . . 11  |-  ( ( B  e.  U  /\  y  =  B )  ->  y  e.  U )
76rexlimivw 2945 . . . . . . . . . 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 3567 . . . . . . 7  |-  ( A. x  e.  A  B  e.  U  ->  { y  |  E. x  e.  A  y  =  B }  C_  U )
113, 10syl5eqss 3541 . . . . . 6  |-  ( A. x  e.  A  B  e.  U  ->  ran  (
x  e.  A  |->  B )  C_  U )
12 df-f 5583 . . . . . 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 9165 . . . . . 6  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  (
x  e.  A  |->  B ) : A --> U )  ->  U. ran  ( x  e.  A  |->  B )  e.  U )
15143expia 1193 . . . . 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 5246 . . . . 5  |-  ( A. x  e.  A  B  e.  U  ->  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B ) )
1817eleq1d 2529 . . . 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 1188 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 968    = wceq 1374    e. wcel 1762   {cab 2445   A.wral 2807   E.wrex 2808    C_ wss 3469   U.cuni 4238   U_ciun 4318    |-> cmpt 4498   ran crn 4993    Fn wfn 5574   -->wf 5575   Univcgru 9157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-map 7412  df-gru 9158
This theorem is referenced by:  gruuni  9167  gruun  9173  gruixp  9176  grur1a  9186
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