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Theorem dfcoll2 44248
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 44247 . 2 (𝐹 Coll 𝐴) = 𝑥𝐴 Scott (𝐹 “ {𝑥})
2 imasng 6104 . . . 4 (𝑥𝐴 → (𝐹 “ {𝑥}) = {𝑦𝑥𝐹𝑦})
32scotteqd 44233 . . 3 (𝑥𝐴 → Scott (𝐹 “ {𝑥}) = Scott {𝑦𝑥𝐹𝑦})
43iuneq2i 5018 . 2 𝑥𝐴 Scott (𝐹 “ {𝑥}) = 𝑥𝐴 Scott {𝑦𝑥𝐹𝑦}
51, 4eqtri 2763 1 (𝐹 Coll 𝐴) = 𝑥𝐴 Scott {𝑦𝑥𝐹𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  {cab 2712  {csn 4631   ciun 4996   class class class wbr 5148  cima 5692  Scott cscott 44231   Coll ccoll 44246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-iun 4998  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-scott 44232  df-coll 44247
This theorem is referenced by:  cpcolld  44254  grucollcld  44256
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