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Theorem iunex 6764
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 2825 . 2  |-  A. x  e.  A  B  e.  _V
4 iunexg 6760 . 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 1767   A.wral 2814   _Vcvv 3113   U_ciun 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596
This theorem is referenced by:  abrexex2  6765  tz9.1  8160  tz9.1c  8161  cplem2  8308  fseqdom  8407  pwsdompw  8584  cfsmolem  8650  ac6c4  8861  konigthlem  8943  alephreg  8957  pwfseqlem4  9040  pwfseqlem5  9041  pwxpndom2  9043  wunex2  9116  wuncval2  9125  inar1  9153  isfunc  15091  dfac14  19882  txcmplem2  19906  cnextfval  20325  dfrtrclrec2  28569  rtrclreclem.refl  28570  rtrclreclem.subset  28571  rtrclreclem.min  28573  colinearex  29315  volsupnfl  29664  heiborlem3  29940  bnj893  33083
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