Theorem dfiun3 5839

Description: Alternate definition of indexed union when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
dfiun3.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
dfiun3	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵)

Proof of Theorem dfiun3

Step	Hyp	Ref	Expression
1		dfiun3g 5837	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
2		dfiun3.1	. . 3 ⊢ 𝐵 ∈ V
3	2	a1i 11	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ V)
4	1, 3	mprg 3154	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵)

Syntax hints:   = wceq 1537   ∈ wcel 2114  Vcvv 3496  ∪ cuni 4840  ∪ ciun 4921   ↦ cmpt 5148  ran crn 5558

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-cnv 5565  df-dm 5567  df-rn 5568

This theorem is referenced by:  tgrest  21769  comppfsc  22142  sigapildsys  31423  ldgenpisyslem1  31424  dstfrvunirn  31734  ctbssinf  34689  mblfinlem2  34932  volsupnfl  34939  istotbnd3  35051  sstotbnd  35055  fourierdlem80  42478