Theorem dfcoll2 41870

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 41869	. 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥})
2		imasng 5991	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝐹 “ {𝑥}) = {𝑦 ∣ 𝑥𝐹𝑦})
3	2	scotteqd 41855	. . 3 ⊢ (𝑥 ∈ 𝐴 → Scott (𝐹 “ {𝑥}) = Scott {𝑦 ∣ 𝑥𝐹𝑦})
4	3	iuneq2i 4945	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥}) = ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦}
5	1, 4	eqtri 2766	1 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦}

Syntax hints:   = wceq 1539   ∈ wcel 2106  {cab 2715  {csn 4561  ∪ ciun 4924   class class class wbr 5074   “ cima 5592  Scott cscott 41853   Coll ccoll 41868

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-iun 4926  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-scott 41854  df-coll 41869

This theorem is referenced by:  cpcolld  41876  grucollcld  41878