Theorem iunexg 7954

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 5033	. . 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 7951	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
4	3	uniexd 7736	. . 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 2832	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  {cab 2708  ∀wral 3060  ∃wrex 3069  Vcvv 3473  ∪ cuni 4908  ∪ ciun 4997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-11 2153  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-mo 2533  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-v 3475  df-in 3955  df-ss 3965  df-uni 4909  df-iun 4999

This theorem is referenced by:  abrexex2g  7955  opabex3d  7956  opabex3rd  7957  opabex3  7958  iunex  7959  xpexgALT  7972  mpoexxg  8066  ixpexg  8922  ixpssmapg  8928  ttrclselem2  9727  iundom  10543  iunctb  10575  wrdexg  14481  cshwsex  17041  imasplusg  17470  imasmulr  17471  imasvsca  17473  imasip  17474  gsum2d2  19890  gsumcom2  19891  dprd2da  19960  ptcls  23441  ptcmplem2  23878  elpwiuncl  32200  aciunf1lem  32322  gsumpart  32645  irngval  33206  esum2dlem  33556  esum2d  33557  esumiun  33558  omssubadd  33765  eulerpartlemgs2  33845  bnj535  34367  bnj546  34373  bnj893  34405  bnj1136  34474  bnj1413  34512  eliunov2  42896  fvmptiunrelexplb0d  42901  fvmptiunrelexplb1d  42903  iunrelexp0  42919  collexd  43482  unirnmapsn  44375  iunmapss  44376  ssmapsn  44377  iunmapsn  44378  sge0iunmptlemfi  45591  sge0iunmpt  45596  smflimlem1  45949  smfliminflem  46008  mpoexxg2  47179