Theorem dfcoll2 43963

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 43962	. 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥})
2		imasng 6085	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝐹 “ {𝑥}) = {𝑦 ∣ 𝑥𝐹𝑦})
3	2	scotteqd 43948	. . 3 ⊢ (𝑥 ∈ 𝐴 → Scott (𝐹 “ {𝑥}) = Scott {𝑦 ∣ 𝑥𝐹𝑦})
4	3	iuneq2i 5014	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥}) = ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦}
5	1, 4	eqtri 2754	1 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦}

Syntax hints:   = wceq 1534   ∈ wcel 2099  {cab 2703  {csn 4623  ∪ ciun 4993   class class class wbr 5145   “ cima 5677  Scott cscott 43946   Coll ccoll 43961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-iun 4995  df-br 5146  df-opab 5208  df-xp 5680  df-cnv 5682  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-scott 43947  df-coll 43962

This theorem is referenced by:  cpcolld  43969  grucollcld  43971