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Theorem dfcoll2 44276
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 44275 . 2 (𝐹 Coll 𝐴) = 𝑥𝐴 Scott (𝐹 “ {𝑥})
2 imasng 6101 . . . 4 (𝑥𝐴 → (𝐹 “ {𝑥}) = {𝑦𝑥𝐹𝑦})
32scotteqd 44261 . . 3 (𝑥𝐴 → Scott (𝐹 “ {𝑥}) = Scott {𝑦𝑥𝐹𝑦})
43iuneq2i 5012 . 2 𝑥𝐴 Scott (𝐹 “ {𝑥}) = 𝑥𝐴 Scott {𝑦𝑥𝐹𝑦}
51, 4eqtri 2764 1 (𝐹 Coll 𝐴) = 𝑥𝐴 Scott {𝑦𝑥𝐹𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2107  {cab 2713  {csn 4625   ciun 4990   class class class wbr 5142  cima 5687  Scott cscott 44259   Coll ccoll 44274
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-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-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-iun 4992  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-scott 44260  df-coll 44275
This theorem is referenced by:  cpcolld  44282  grucollcld  44284
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