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Theorem dfiin3 5584
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 5582 . 2 (∀𝑥𝐴 𝐵 ∈ V → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
2 dfiun3.1 . . 3 𝐵 ∈ V
32a1i 11 . 2 (𝑥𝐴𝐵 ∈ V)
41, 3mprg 3106 1 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  wcel 2157  Vcvv 3384   cint 4666   ciin 4710  cmpt 4921  ran crn 5312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-sep 4974  ax-nul 4982  ax-pr 5096
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3386  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-int 4667  df-iin 4712  df-br 4843  df-opab 4905  df-mpt 4922  df-cnv 5319  df-dm 5321  df-rn 5322
This theorem is referenced by:  fclscmpi  22158
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