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Theorem compss 9081
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 9076 . . 3 𝐹 = 𝐹
32imaeq1i 5382 . 2 (𝐹𝐺) = (𝐹𝐺)
4 difeq2 3684 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
54cbvmptv 4678 . . . 4 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴𝑦))
61, 5eqtri 2632 . . 3 𝐹 = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴𝑦))
76mptpreima 5545 . 2 (𝐹𝐺) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝐴𝑦) ∈ 𝐺}
83, 7eqtr3i 2634 1 (𝐹𝐺) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝐴𝑦) ∈ 𝐺}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  wcel 1977  {crab 2900  cdif 3537  𝒫 cpw 4108  cmpt 4643  ccnv 5037  cima 5041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051
This theorem is referenced by:  isf34lem4  9082
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