Theorem dfcoll2 44493

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 44492	. 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥})
2		imasng 6043	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝐹 “ {𝑥}) = {𝑦 ∣ 𝑥𝐹𝑦})
3	2	scotteqd 44478	. . 3 ⊢ (𝑥 ∈ 𝐴 → Scott (𝐹 “ {𝑥}) = Scott {𝑦 ∣ 𝑥𝐹𝑦})
4	3	iuneq2i 4968	. 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 1541   ∈ wcel 2113  {cab 2714  {csn 4580  ∪ ciun 4946   class class class wbr 5098   “ cima 5627  Scott cscott 44476   Coll ccoll 44491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-iun 4948  df-br 5099  df-opab 5161  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-scott 44477  df-coll 44492

This theorem is referenced by:  cpcolld  44499  grucollcld  44501