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Theorem iunex 7347
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 ∈ V
iunex.2 𝐵 ∈ V
Assertion
Ref Expression
iunex 𝑥𝐴 𝐵 ∈ V
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunex
StepHypRef Expression
1 iunex.1 . 2 𝐴 ∈ V
2 iunex.2 . . 3 𝐵 ∈ V
32rgenw 3071 . 2 𝑥𝐴 𝐵 ∈ V
4 iunexg 7343 . 2 ((𝐴 ∈ V ∧ ∀𝑥𝐴 𝐵 ∈ V) → 𝑥𝐴 𝐵 ∈ V)
51, 3, 4mp2an 683 1 𝑥𝐴 𝐵 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2155  wral 3055  Vcvv 3350   ciun 4678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078
This theorem is referenced by:  abrexex2OLD  7350  tz9.1  8822  tz9.1c  8823  cplem2  8970  fseqdom  9102  pwsdompw  9281  cfsmolem  9347  ac6c4  9558  konigthlem  9645  alephreg  9659  pwfseqlem4  9739  pwfseqlem5  9740  pwxpndom2  9742  wunex2  9815  wuncval2  9824  inar1  9852  dfrtrclrec2  14085  rtrclreclem1  14086  rtrclreclem2  14087  rtrclreclem4  14089  isfunc  16792  dfac14  21704  txcmplem2  21728  cnextfval  22148  bnj893  31449  colinearex  32614  volsupnfl  33881  heiborlem3  34037  comptiunov2i  38676  corclrcl  38677  iunrelexpmin1  38678  trclrelexplem  38681  iunrelexpmin2  38682  dftrcl3  38690  trclfvcom  38693  cnvtrclfv  38694  cotrcltrcl  38695  trclimalb2  38696  trclfvdecomr  38698  dfrtrcl3  38703  dfrtrcl4  38708  corcltrcl  38709  cotrclrcl  38712  carageniuncllem1  41378  carageniuncllem2  41379  carageniuncl  41380  caratheodorylem1  41383  caratheodorylem2  41384  ovnovollem1  41513  ovnovollem2  41514  smfresal  41638
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