Mathbox for Rohan Ridenour < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfcoll2 Structured version   Visualization version   GIF version

Theorem dfcoll2 40880
 Description: Alternate definition of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Assertion
Ref Expression
dfcoll2 (𝐹 Coll 𝐴) = 𝑥𝐴 Scott {𝑦𝑥𝐹𝑦}
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐹
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem dfcoll2
StepHypRef Expression
1 df-coll 40879 . 2 (𝐹 Coll 𝐴) = 𝑥𝐴 Scott (𝐹 “ {𝑥})
2 imasng 5938 . . . 4 (𝑥𝐴 → (𝐹 “ {𝑥}) = {𝑦𝑥𝐹𝑦})
32scotteqd 40865 . . 3 (𝑥𝐴 → Scott (𝐹 “ {𝑥}) = Scott {𝑦𝑥𝐹𝑦})
43iuneq2i 4926 . 2 𝑥𝐴 Scott (𝐹 “ {𝑥}) = 𝑥𝐴 Scott {𝑦𝑥𝐹𝑦}
51, 4eqtri 2847 1 (𝐹 Coll 𝐴) = 𝑥𝐴 Scott {𝑦𝑥𝐹𝑦}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2115  {cab 2802  {csn 4550  ∪ ciun 4905   class class class wbr 5052   “ cima 5545  Scott cscott 40863   Coll ccoll 40878 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-iun 4907  df-br 5053  df-opab 5115  df-xp 5548  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-scott 40864  df-coll 40879 This theorem is referenced by:  cpcolld  40886  grucollcld  40888
 Copyright terms: Public domain W3C validator