MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfiin3g Structured version   Visualization version   GIF version

Theorem dfiin3g 5534
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 4705 . 2 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
2 eqid 2760 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32rnmpt 5526 . . 3 ran (𝑥𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
43inteqi 4631 . 2 ran (𝑥𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
51, 4syl6eqr 2812 1 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  {cab 2746  wral 3050  wrex 3051   cint 4627   ciin 4673  cmpt 4881  ran crn 5267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-int 4628  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-cnv 5274  df-dm 5276  df-rn 5277
This theorem is referenced by:  dfiin3  5536  riinint  5537  iinon  7607  cmpfi  21433
  Copyright terms: Public domain W3C validator