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Theorem iunex 6662
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 2895 . 2  |-  A. x  e.  A  B  e.  _V
4 iunexg 6658 . 2  |-  ( ( A  e.  _V  /\  A. x  e.  A  B  e.  _V )  ->  U_ x  e.  A  B  e.  _V )
51, 3, 4mp2an 672 1  |-  U_ x  e.  A  B  e.  _V
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1758   A.wral 2796   _Vcvv 3072   U_ciun 4274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529
This theorem is referenced by:  abrexex2  6663  tz9.1  8055  tz9.1c  8056  cplem2  8203  fseqdom  8302  pwsdompw  8479  cfsmolem  8545  ac6c4  8756  konigthlem  8838  alephreg  8852  pwfseqlem4  8935  pwfseqlem5  8936  pwxpndom2  8938  wunex2  9011  wuncval2  9020  inar1  9048  isfunc  14888  dfac14  19318  txcmplem2  19342  cnextfval  19761  dfrtrclrec2  27484  rtrclreclem.refl  27485  rtrclreclem.subset  27486  rtrclreclem.min  27488  colinearex  28230  volsupnfl  28579  heiborlem3  28855  bnj893  32234
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