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  23440  ptcmplem2  23877  elpwiuncl  32198  aciunf1lem  32320  gsumpart  32643  irngval  33204  esum2dlem  33554  esum2d  33555  esumiun  33556  omssubadd  33763  eulerpartlemgs2  33843  bnj535  34365  bnj546  34371  bnj893  34403  bnj1136  34472  bnj1413  34510  eliunov2  42893  fvmptiunrelexplb0d  42898  fvmptiunrelexplb1d  42900  iunrelexp0  42916  collexd  43479  unirnmapsn  44372  iunmapss  44373  ssmapsn  44374  iunmapsn  44375  sge0iunmptlemfi  45588  sge0iunmpt  45593  smflimlem1  45946  smfliminflem  46005  mpoexxg2  47176