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Theorem iunex 7039
 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 2908 . 2 𝑥𝐴 𝐵 ∈ V
4 iunexg 7035 . 2 ((𝐴 ∈ V ∧ ∀𝑥𝐴 𝐵 ∈ V) → 𝑥𝐴 𝐵 ∈ V)
51, 3, 4mp2an 704 1 𝑥𝐴 𝐵 ∈ V
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ∪ ciun 4455 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812 This theorem is referenced by:  abrexex2  7040  tz9.1  8488  tz9.1c  8489  cplem2  8636  fseqdom  8732  pwsdompw  8909  cfsmolem  8975  ac6c4  9186  konigthlem  9269  alephreg  9283  pwfseqlem4  9363  pwfseqlem5  9364  pwxpndom2  9366  wunex2  9439  wuncval2  9448  inar1  9476  dfrtrclrec2  13645  rtrclreclem1  13646  rtrclreclem2  13647  rtrclreclem4  13649  isfunc  16347  dfac14  21231  txcmplem2  21255  cnextfval  21676  bnj893  30252  colinearex  31337  volsupnfl  32624  heiborlem3  32782  comptiunov2i  37017  corclrcl  37018  iunrelexpmin1  37019  trclrelexplem  37022  iunrelexpmin2  37023  dftrcl3  37031  trclfvcom  37034  cnvtrclfv  37035  cotrcltrcl  37036  trclimalb2  37037  trclfvdecomr  37039  dfrtrcl3  37044  dfrtrcl4  37049  corcltrcl  37050  cotrclrcl  37053  carageniuncllem1  39411  carageniuncllem2  39412  carageniuncl  39413  caratheodorylem1  39416  caratheodorylem2  39417  ovnovollem1  39546  ovnovollem2  39547  smfresal  39673
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