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Theorem iunex 7189
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 2953 . 2 𝑥𝐴 𝐵 ∈ V
4 iunexg 7185 . 2 ((𝐴 ∈ V ∧ ∀𝑥𝐴 𝐵 ∈ V) → 𝑥𝐴 𝐵 ∈ V)
51, 3, 4mp2an 708 1 𝑥𝐴 𝐵 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2030  wral 2941  Vcvv 3231   ciun 4552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934
This theorem is referenced by:  abrexex2OLD  7192  tz9.1  8643  tz9.1c  8644  cplem2  8791  fseqdom  8887  pwsdompw  9064  cfsmolem  9130  ac6c4  9341  konigthlem  9428  alephreg  9442  pwfseqlem4  9522  pwfseqlem5  9523  pwxpndom2  9525  wunex2  9598  wuncval2  9607  inar1  9635  dfrtrclrec2  13841  rtrclreclem1  13842  rtrclreclem2  13843  rtrclreclem4  13845  isfunc  16571  dfac14  21469  txcmplem2  21493  cnextfval  21913  bnj893  31124  colinearex  32292  volsupnfl  33584  heiborlem3  33742  comptiunov2i  38315  corclrcl  38316  iunrelexpmin1  38317  trclrelexplem  38320  iunrelexpmin2  38321  dftrcl3  38329  trclfvcom  38332  cnvtrclfv  38333  cotrcltrcl  38334  trclimalb2  38335  trclfvdecomr  38337  dfrtrcl3  38342  dfrtrcl4  38347  corcltrcl  38348  cotrclrcl  38351  carageniuncllem1  41056  carageniuncllem2  41057  carageniuncl  41058  caratheodorylem1  41061  caratheodorylem2  41062  ovnovollem1  41191  ovnovollem2  41192  smfresal  41316
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