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Theorem iunex 5991
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 2773 . 2  |-  A. x  e.  A  B  e.  _V
4 iunexg 5987 . 2  |-  ( ( A  e.  _V  /\  A. x  e.  A  B  e.  _V )  ->  U_ x  e.  A  B  e.  _V )
51, 3, 4mp2an 654 1  |-  U_ x  e.  A  B  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   A.wral 2705   _Vcvv 2956   U_ciun 4093
This theorem is referenced by:  abrexex2  6001  tz9.1  7665  tz9.1c  7666  cplem2  7814  fseqdom  7907  pwsdompw  8084  cfsmolem  8150  ac6c4  8361  konigthlem  8443  alephreg  8457  pwfseqlem4  8537  pwfseqlem5  8538  pwxpndom2  8540  wunex2  8613  wuncval2  8622  inar1  8650  isfunc  14061  dfac14  17650  txcmplem2  17674  cnextfval  18093  dfrtrclrec2  25143  rtrclreclem.refl  25144  rtrclreclem.subset  25145  rtrclreclem.min  25147  colinearex  25994  volsupnfl  26251  heiborlem3  26522  bnj893  29299
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462
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