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Theorem iunex 7661
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 3148 . 2 𝑥𝐴 𝐵 ∈ V
4 iunexg 7656 . 2 ((𝐴 ∈ V ∧ ∀𝑥𝐴 𝐵 ∈ V) → 𝑥𝐴 𝐵 ∈ V)
51, 3, 4mp2an 690 1 𝑥𝐴 𝐵 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  wral 3136  Vcvv 3493   ciun 4910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356
This theorem is referenced by:  tz9.1  9163  tz9.1c  9164  cplem2  9311  fseqdom  9444  pwsdompw  9618  cfsmolem  9684  ac6c4  9895  konigthlem  9982  alephreg  9996  pwfseqlem4  10076  pwfseqlem5  10077  pwxpndom2  10079  wunex2  10152  wuncval2  10161  inar1  10189  dfrtrclrec2  14408  rtrclreclem1  14409  rtrclreclem2  14410  rtrclreclem4  14412  isfunc  17126  smndex1bas  18063  smndex1sgrp  18065  smndex1mnd  18067  smndex1id  18068  dfac14  22218  txcmplem2  22242  cnextfval  22662  bnj893  32193  colinearex  33514  volsupnfl  34929  heiborlem3  35083  comptiunov2i  40041  corclrcl  40042  iunrelexpmin1  40043  trclrelexplem  40046  iunrelexpmin2  40047  dftrcl3  40055  trclfvcom  40058  cnvtrclfv  40059  cotrcltrcl  40060  trclimalb2  40061  trclfvdecomr  40063  dfrtrcl3  40068  dfrtrcl4  40073  corcltrcl  40074  cotrclrcl  40077  carageniuncllem1  42793  carageniuncllem2  42794  carageniuncl  42795  caratheodorylem1  42798  caratheodorylem2  42799  ovnovollem1  42928  ovnovollem2  42929  smfresal  43053
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