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Theorem gruiun 8437
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's 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 2296 . . . . . . 7  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
21fnmpt 5386 . . . . . 6  |-  ( A. x  e.  A  B  e.  U  ->  ( x  e.  A  |->  B )  Fn  A )
31rnmpt 4941 . . . . . . 7  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
4 r19.29 2696 . . . . . . . . . 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 2356 . . . . . . . . . . . 12  |-  ( y  =  B  ->  (
y  e.  U  <->  B  e.  U ) )
65biimparc 473 . . . . . . . . . . 11  |-  ( ( B  e.  U  /\  y  =  B )  ->  y  e.  U )
76rexlimivw 2676 . . . . . . . . . 10  |-  ( E. x  e.  A  ( B  e.  U  /\  y  =  B )  ->  y  e.  U )
84, 7syl 15 . . . . . . . . 9  |-  ( ( A. x  e.  A  B  e.  U  /\  E. x  e.  A  y  =  B )  -> 
y  e.  U )
98ex 423 . . . . . . . 8  |-  ( A. x  e.  A  B  e.  U  ->  ( E. x  e.  A  y  =  B  ->  y  e.  U ) )
109abssdv 3260 . . . . . . 7  |-  ( A. x  e.  A  B  e.  U  ->  { y  |  E. x  e.  A  y  =  B }  C_  U )
113, 10syl5eqss 3235 . . . . . 6  |-  ( A. x  e.  A  B  e.  U  ->  ran  (
x  e.  A  |->  B )  C_  U )
12 df-f 5275 . . . . . 6  |-  ( ( x  e.  A  |->  B ) : A --> U  <->  ( (
x  e.  A  |->  B )  Fn  A  /\  ran  ( x  e.  A  |->  B )  C_  U
) )
132, 11, 12sylanbrc 645 . . . . 5  |-  ( A. x  e.  A  B  e.  U  ->  ( x  e.  A  |->  B ) : A --> U )
14 gruurn 8436 . . . . . 6  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  (
x  e.  A  |->  B ) : A --> U )  ->  U. ran  ( x  e.  A  |->  B )  e.  U )
15143expia 1153 . . . . 5  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( x  e.  A  |->  B ) : A --> U  ->  U. ran  ( x  e.  A  |->  B )  e.  U ) )
1613, 15syl5com 26 . . . 4  |-  ( A. x  e.  A  B  e.  U  ->  ( ( U  e.  Univ  /\  A  e.  U )  ->  U. ran  ( x  e.  A  |->  B )  e.  U
) )
17 dfiun3g 4947 . . . . 5  |-  ( A. x  e.  A  B  e.  U  ->  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B ) )
1817eleq1d 2362 . . . 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 225 . . 3  |-  ( A. x  e.  A  B  e.  U  ->  ( ( U  e.  Univ  /\  A  e.  U )  ->  U_ x  e.  A  B  e.  U ) )
2019com12 27 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( A. x  e.  A  B  e.  U  ->  U_ x  e.  A  B  e.  U ) )
21203impia 1148 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557    C_ wss 3165   U.cuni 3843   U_ciun 3921    e. cmpt 4093   ran crn 4706    Fn wfn 5266   -->wf 5267   Univcgru 8428
This theorem is referenced by:  gruuni  8438  gruun  8444  gruixp  8447  grur1a  8457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-gru 8429
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