Theorem dfiun2 5056

Description: Alternate definition of indexed union when 𝐵 is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 27-Jun-1998.) (Revised by David Abernethy, 19-Jun-2012.)

Hypothesis

Ref	Expression
dfiun2.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
dfiun2	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem dfiun2

Step	Hyp	Ref	Expression
1		dfiun2g 5053	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
2		dfiun2.1	. . 3 ⊢ 𝐵 ∈ V
3	2	a1i 11	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ V)
4	1, 3	mprg 3073	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}

Syntax hints:   = wceq 1537   ∈ wcel 2108  {cab 2717  ∃wrex 3076  Vcvv 3488  ∪ cuni 4931  ∪ ciun 5015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-v 3490  df-uni 4932  df-iun 5017

This theorem is referenced by:  fniunfv  7284  funcnvuni  7972  fiun  7983  f1iun  7984  tfrlem8  8440  rdglim2a  8489  rankuni  9932  cardiun  10051  kmlem11  10230  cfslb2n  10337  enfin2i  10390  pwcfsdom  10652  rankcf  10846  tskuni  10852  discmp  23427  cmpsublem  23428  cmpsub  23429  nnoeomeqom  43274