Theorem dfcoll2 42200

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 42199	. 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥})
2		imasng 6021	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝐹 “ {𝑥}) = {𝑦 ∣ 𝑥𝐹𝑦})
3	2	scotteqd 42185	. . 3 ⊢ (𝑥 ∈ 𝐴 → Scott (𝐹 “ {𝑥}) = Scott {𝑦 ∣ 𝑥𝐹𝑦})
4	3	iuneq2i 4962	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥}) = ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦}
5	1, 4	eqtri 2764	1 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦}

Syntax hints:   = wceq 1540   ∈ wcel 2105  {cab 2713  {csn 4573  ∪ ciun 4941   class class class wbr 5092   “ cima 5623  Scott cscott 42183   Coll ccoll 42198

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-iun 4943  df-br 5093  df-opab 5155  df-xp 5626  df-cnv 5628  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-scott 42184  df-coll 42199

This theorem is referenced by:  cpcolld  42206  grucollcld  42208