Theorem iunex 7973

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	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunex

Step	Hyp	Ref	Expression
1		iunex.1	. 2 ⊢ 𝐴 ∈ V
2		iunex.2	. . 3 ⊢ 𝐵 ∈ V
3	2	rgenw 3054	. 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ V
4		iunexg 7968	. 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ V) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V)
5	1, 3, 4	mp2an 690	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V

Syntax hints:   ∈ wcel 2098  ∀wral 3050  Vcvv 3461  ∪ ciun 4997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-mo 2528  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-v 3463  df-ss 3961  df-uni 4910  df-iun 4999

This theorem is referenced by:  tz9.1  9754  tz9.1c  9755  cplem2  9915  fseqdom  10051  pwsdompw  10229  cfsmolem  10295  ac6c4  10506  konigthlem  10593  alephreg  10607  pwfseqlem4  10687  pwfseqlem5  10688  pwxpndom2  10690  wunex2  10763  wuncval2  10772  inar1  10800  rtrclreclem1  15040  dfrtrclrec2  15041  rtrclreclem2  15042  rtrclreclem4  15044  isfunc  17853  smndex1bas  18866  smndex1sgrp  18868  smndex1mnd  18870  smndex1id  18871  dfac14  23566  txcmplem2  23590  cnextfval  24010  bnj893  34690  colinearex  35787  volsupnfl  37269  heiborlem3  37417  comptiunov2i  43278  corclrcl  43279  iunrelexpmin1  43280  trclrelexplem  43283  iunrelexpmin2  43284  dftrcl3  43292  trclfvcom  43295  cnvtrclfv  43296  cotrcltrcl  43297  trclimalb2  43298  trclfvdecomr  43300  dfrtrcl3  43305  dfrtrcl4  43310  corcltrcl  43311  cotrclrcl  43314  carageniuncllem1  46047  carageniuncllem2  46048  carageniuncl  46049  caratheodorylem1  46052  caratheodorylem2  46053  ovnovollem1  46182  ovnovollem2  46183  smfresal  46314