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Theorem dfcoll2 42624
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 42623 . 2 (𝐹 Coll 𝐴) = 𝑥𝐴 Scott (𝐹 “ {𝑥})
2 imasng 6039 . . . 4 (𝑥𝐴 → (𝐹 “ {𝑥}) = {𝑦𝑥𝐹𝑦})
32scotteqd 42609 . . 3 (𝑥𝐴 → Scott (𝐹 “ {𝑥}) = Scott {𝑦𝑥𝐹𝑦})
43iuneq2i 4979 . 2 𝑥𝐴 Scott (𝐹 “ {𝑥}) = 𝑥𝐴 Scott {𝑦𝑥𝐹𝑦}
51, 4eqtri 2761 1 (𝐹 Coll 𝐴) = 𝑥𝐴 Scott {𝑦𝑥𝐹𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  {cab 2710  {csn 4590   ciun 4958   class class class wbr 5109  cima 5640  Scott cscott 42607   Coll ccoll 42622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-iun 4960  df-br 5110  df-opab 5172  df-xp 5643  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-scott 42608  df-coll 42623
This theorem is referenced by:  cpcolld  42630  grucollcld  42632
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