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Theorem dfiin3 5807
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 5805 . 2 (∀𝑥𝐴 𝐵 ∈ V → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
2 dfiun3.1 . . 3 𝐵 ∈ V
32a1i 11 . 2 (𝑥𝐴𝐵 ∈ V)
41, 3mprg 3123 1 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2112  Vcvv 3444   cint 4841   ciin 4885  cmpt 5113  ran crn 5524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-int 4842  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-cnv 5531  df-dm 5533  df-rn 5534
This theorem is referenced by:  subdrgint  19579  fclscmpi  22638
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