Theorem iunexg 7921

Description: The existence of an indexed union. 𝑥 is normally a free-variable parameter in 𝐵. (Contributed by NM, 23-Mar-2006.)

Assertion

Ref	Expression
iunexg	⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem iunexg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiun2g 4990	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
2	1	adantl 481	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
3		abrexexg 7919	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
4	3	uniexd 7698	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
5	4	adantr 480	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
6	2, 5	eqeltrd 2828	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053  Vcvv 3444  ∪ cuni 4867  ∪ ciun 4951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-v 3446  df-ss 3928  df-uni 4868  df-iun 4953

This theorem is referenced by:  abrexex2g  7922  opabex3d  7923  opabex3rd  7924  opabex3  7925  iunex  7926  xpexgALT  7939  mpoexxg  8033  ixpexg  8872  ixpssmapg  8878  ttrclselem2  9655  iundom  10471  iunctb  10503  wrdexg  14465  cshwsex  17047  imasplusg  17456  imasmulr  17457  imasvsca  17459  imasip  17460  gsum2d2  19888  gsumcom2  19889  dprd2da  19958  ptcls  23536  ptcmplem2  23973  elpwiuncl  32506  aciunf1lem  32636  gsumpart  33040  gsumwrd2dccat  33050  irngval  33673  esum2dlem  34075  esum2d  34076  esumiun  34077  omssubadd  34284  eulerpartlemgs2  34364  bnj535  34873  bnj546  34879  bnj893  34911  bnj1136  34980  bnj1413  35018  weiunse  36449  numiunnum  36451  eliunov2  43661  fvmptiunrelexplb0d  43666  fvmptiunrelexplb1d  43668  iunrelexp0  43684  collexd  44239  unirnmapsn  45201  iunmapss  45202  ssmapsn  45203  iunmapsn  45204  sge0iunmptlemfi  46404  sge0iunmpt  46409  smflimlem1  46762  smfliminflem  46821  mpoexxg2  48319  imasubclem1  49086