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Theorem dfiin3g 5978
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 5031 . 2 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
2 eqid 2736 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32rnmpt 5967 . . 3 ran (𝑥𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
43inteqi 4949 . 2 ran (𝑥𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
51, 4eqtr4di 2794 1 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  {cab 2713  wral 3060  wrex 3069   cint 4945   ciin 4991  cmpt 5224  ran crn 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-int 4946  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-cnv 5692  df-dm 5694  df-rn 5695
This theorem is referenced by:  dfiin3  5980  riinint  5981  iinon  8381  cmpfi  23417
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