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Theorem compss 9786
Description: Express image under of the complementation isomorphism. (Contributed by Stefan O'Rear, 5-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
Hypothesis
Ref Expression
compss.a 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
Assertion
Ref Expression
compss (𝐹𝐺) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝐴𝑦) ∈ 𝐺}
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐹   𝑦,𝐺
Allowed substitution hints:   𝐹(𝑥)   𝐺(𝑥)

Proof of Theorem compss
StepHypRef Expression
1 compss.a . . . 4 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
21compsscnv 9781 . . 3 𝐹 = 𝐹
32imaeq1i 5919 . 2 (𝐹𝐺) = (𝐹𝐺)
4 difeq2 4090 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
54cbvmptv 5160 . . . 4 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴𝑦))
61, 5eqtri 2841 . . 3 𝐹 = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴𝑦))
76mptpreima 6085 . 2 (𝐹𝐺) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝐴𝑦) ∈ 𝐺}
83, 7eqtr3i 2843 1 (𝐹𝐺) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝐴𝑦) ∈ 𝐺}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  {crab 3139  cdif 3930  𝒫 cpw 4535  cmpt 5137  ccnv 5547  cima 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-mpt 5138  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by:  isf34lem4  9787
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