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Theorem dfiin3 5302
 Description: Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
dfiun3.1 𝐵 ∈ V
Assertion
Ref Expression
dfiin3 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)

Proof of Theorem dfiin3
StepHypRef Expression
1 dfiin3g 5300 . 2 (∀𝑥𝐴 𝐵 ∈ V → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
2 dfiun3.1 . . 3 𝐵 ∈ V
32a1i 11 . 2 (𝑥𝐴𝐵 ∈ V)
41, 3mprg 2910 1 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∩ cint 4410  ∩ ciin 4456   ↦ cmpt 4643  ran crn 5039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-int 4411  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-cnv 5046  df-dm 5048  df-rn 5049 This theorem is referenced by:  fclscmpi  21643
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