Theorem dfiun3 5969

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 5967	. 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 3056	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵)

Syntax hints:   = wceq 1533   ∈ wcel 2098  Vcvv 3461  ∪ cuni 4909  ∪ ciun 4997   ↦ cmpt 5232  ran crn 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-cnv 5686  df-dm 5688  df-rn 5689

This theorem is referenced by:  tgrest  23107  comppfsc  23480  sigapildsys  33912  ldgenpisyslem1  33913  dstfrvunirn  34225  ctbssinf  37016  mblfinlem2  37262  volsupnfl  37269  istotbnd3  37375  sstotbnd  37379  rp-tfslim  42924  fourierdlem80  45712