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Theorem iunex 5786
Description: The existence of an indexed union.  x is normally a free-variable parameter in the class expression substituted for  B, which can be read informally as  B ( x ). (Contributed by NM, 13-Oct-2003.)
Hypotheses
Ref Expression
iunex.1  |-  A  e. 
_V
iunex.2  |-  B  e. 
_V
Assertion
Ref Expression
iunex  |-  U_ x  e.  A  B  e.  _V
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem iunex
StepHypRef Expression
1 iunex.1 . 2  |-  A  e. 
_V
2 iunex.2 . . 3  |-  B  e. 
_V
32rgenw 2623 . 2  |-  A. x  e.  A  B  e.  _V
4 iunexg 5783 . 2  |-  ( ( A  e.  _V  /\  A. x  e.  A  B  e.  _V )  ->  U_ x  e.  A  B  e.  _V )
51, 3, 4mp2an 653 1  |-  U_ x  e.  A  B  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1696   A.wral 2556   _Vcvv 2801   U_ciun 3921
This theorem is referenced by:  abrexex2  5796  tz9.1  7427  tz9.1c  7428  cplem2  7576  fseqdom  7669  pwsdompw  7846  cfsmolem  7912  ac6c4  8124  konigthlem  8206  alephreg  8220  pwfseqlem4  8300  pwfseqlem5  8301  pwxpndom2  8303  wunex2  8376  wuncval2  8385  inar1  8413  isfunc  13754  dfac14  17328  txcmplem2  17352  dfrtrclrec2  24055  rtrclreclem.refl  24056  rtrclreclem.subset  24057  rtrclreclem.min  24059  colinearex  24755  trclval  25997  heiborlem3  26640  bnj893  29276
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-rep 4147  ax-sep 4157  ax-nul 4165  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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  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-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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