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Theorem gruiun 8421
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 2283 . . . . . . 7  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
21fnmpt 5370 . . . . . 6  |-  ( A. x  e.  A  B  e.  U  ->  ( x  e.  A  |->  B )  Fn  A )
31rnmpt 4925 . . . . . . 7  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
4 r19.29 2683 . . . . . . . . . 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 2343 . . . . . . . . . . . 12  |-  ( y  =  B  ->  (
y  e.  U  <->  B  e.  U ) )
65biimparc 473 . . . . . . . . . . 11  |-  ( ( B  e.  U  /\  y  =  B )  ->  y  e.  U )
76rexlimivw 2663 . . . . . . . . . 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 3247 . . . . . . 7  |-  ( A. x  e.  A  B  e.  U  ->  { y  |  E. x  e.  A  y  =  B }  C_  U )
113, 10syl5eqss 3222 . . . . . 6  |-  ( A. x  e.  A  B  e.  U  ->  ran  (
x  e.  A  |->  B )  C_  U )
12 df-f 5259 . . . . . 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 8420 . . . . . 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 4931 . . . . 5  |-  ( A. x  e.  A  B  e.  U  ->  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B ) )
1817eleq1d 2349 . . . 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 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544    C_ wss 3152   U.cuni 3827   U_ciun 3905    e. cmpt 4077   ran crn 4690    Fn wfn 5250   -->wf 5251   Univcgru 8412
This theorem is referenced by:  gruuni  8422  gruun  8428  gruixp  8431  grur1a  8441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-gru 8413
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