Theorem dfcoll2 44243

Description: Alternate definition of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)

Assertion

Ref	Expression
dfcoll2	⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦}

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐹

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem dfcoll2

Step	Hyp	Ref	Expression
1		df-coll 44242	. 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥})
2		imasng 6076	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝐹 “ {𝑥}) = {𝑦 ∣ 𝑥𝐹𝑦})
3	2	scotteqd 44228	. . 3 ⊢ (𝑥 ∈ 𝐴 → Scott (𝐹 “ {𝑥}) = Scott {𝑦 ∣ 𝑥𝐹𝑦})
4	3	iuneq2i 4994	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥}) = ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦}
5	1, 4	eqtri 2759	1 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦}

Syntax hints:   = wceq 1540   ∈ wcel 2109  {cab 2714  {csn 4606  ∪ ciun 4972   class class class wbr 5124   “ cima 5662  Scott cscott 44226   Coll ccoll 44241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-iun 4974  df-br 5125  df-opab 5187  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-scott 44227  df-coll 44242

This theorem is referenced by:  cpcolld  44249  grucollcld  44251