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Theorem dfiun3 5958
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 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)

Proof of Theorem dfiun3
StepHypRef Expression
1 dfiun3g 5956 . 2 (∀𝑥𝐴 𝐵 ∈ V → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
2 dfiun3.1 . . 3 𝐵 ∈ V
32a1i 11 . 2 (𝑥𝐴𝐵 ∈ V)
41, 3mprg 3061 1 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  Vcvv 3468   cuni 4902   ciun 4990  cmpt 5224  ran crn 5670
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-cnv 5677  df-dm 5679  df-rn 5680
This theorem is referenced by:  tgrest  23014  comppfsc  23387  sigapildsys  33690  ldgenpisyslem1  33691  dstfrvunirn  34003  ctbssinf  36794  mblfinlem2  37037  volsupnfl  37044  istotbnd3  37150  sstotbnd  37154  rp-tfslim  42660  fourierdlem80  45455
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