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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfcoll2 | Structured version Visualization version GIF version |
Description: Alternate definition of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
Ref | Expression |
---|---|
dfcoll2 | ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-coll 44247 | . 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥}) | |
2 | imasng 6104 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝐹 “ {𝑥}) = {𝑦 ∣ 𝑥𝐹𝑦}) | |
3 | 2 | scotteqd 44233 | . . 3 ⊢ (𝑥 ∈ 𝐴 → Scott (𝐹 “ {𝑥}) = Scott {𝑦 ∣ 𝑥𝐹𝑦}) |
4 | 3 | iuneq2i 5018 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥}) = ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦} |
5 | 1, 4 | eqtri 2763 | 1 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 {cab 2712 {csn 4631 ∪ ciun 4996 class class class wbr 5148 “ cima 5692 Scott cscott 44231 Coll ccoll 44246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-iun 4998 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-scott 44232 df-coll 44247 |
This theorem is referenced by: cpcolld 44254 grucollcld 44256 |
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