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Theorem dfiin3g 5834
Description: Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
dfiin3g (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))

Proof of Theorem dfiin3g
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfiin2g 4941 . 2 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
2 eqid 2737 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32rnmpt 5824 . . 3 ran (𝑥𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
43inteqi 4863 . 2 ran (𝑥𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
51, 4eqtr4di 2796 1 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  {cab 2714  wral 3061  wrex 3062   cint 4859   ciin 4905  cmpt 5135  ran crn 5552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-int 4860  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-cnv 5559  df-dm 5561  df-rn 5562
This theorem is referenced by:  dfiin3  5836  riinint  5837  iinon  8077  cmpfi  22305
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