Theorem iunexg 7916

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 4972	. . 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 7914	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
4	3	uniexd 7696	. . 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 2836	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2714  ∀wral 3051  ∃wrex 3061  Vcvv 3429  ∪ cuni 4850  ∪ ciun 4933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-v 3431  df-ss 3906  df-uni 4851  df-iun 4935

This theorem is referenced by:  abrexex2g  7917  opabex3d  7918  opabex3rd  7919  opabex3  7920  iunex  7921  xpexgALT  7934  mpoexxg  8028  ixpexg  8870  ixpssmapg  8876  ttrclselem2  9647  iundom  10464  iunctb  10497  wrdexg  14486  cshwsex  17071  imasplusg  17481  imasmulr  17482  imasvsca  17484  imasip  17485  gsum2d2  19949  gsumcom2  19950  dprd2da  20019  ptcls  23581  ptcmplem2  24018  elpwiuncl  32597  aciunf1lem  32735  gsumpart  33124  gsumwrd2dccat  33139  irngval  33829  esum2dlem  34236  esum2d  34237  esumiun  34238  omssubadd  34444  eulerpartlemgs2  34524  bnj535  35032  bnj546  35038  bnj893  35070  bnj1136  35139  bnj1413  35177  tz9.1regs  35278  weiunse  36650  numiunnum  36652  eliunov2  44106  fvmptiunrelexplb0d  44111  fvmptiunrelexplb1d  44113  iunrelexp0  44129  collexd  44684  unirnmapsn  45643  iunmapss  45644  ssmapsn  45645  iunmapsn  45646  sge0iunmptlemfi  46841  sge0iunmpt  46846  smflimlem1  47199  smfliminflem  47258  mpoexxg2  48814  imasubclem1  49579