Theorem compss 10350

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 ↦ (𝐴 ∖ 𝑥))

Assertion

Ref	Expression
compss	⊢ (𝐹 “ 𝐺) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑦) ∈ 𝐺}

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐹   𝑦,𝐺

Allowed substitution hints:   𝐹(𝑥)   𝐺(𝑥)

Proof of Theorem compss

Step	Hyp	Ref	Expression
1		compss.a	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))
2	1	compsscnv 10345	. . 3 ⊢ ◡𝐹 = 𝐹
3	2	imaeq1i 6043	. 2 ⊢ (◡𝐹 “ 𝐺) = (𝐹 “ 𝐺)
4		difeq2 4109	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
5	4	cbvmptv 5251	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑦))
6	1, 5	eqtri 2759	. . 3 ⊢ 𝐹 = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑦))
7	6	mptpreima 6223	. 2 ⊢ (◡𝐹 “ 𝐺) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑦) ∈ 𝐺}
8	3, 7	eqtr3i 2761	1 ⊢ (𝐹 “ 𝐺) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑦) ∈ 𝐺}

Syntax hints:   = wceq 1541   ∈ wcel 2106  {crab 3429   ∖ cdif 3938  𝒫 cpw 4593   ↦ cmpt 5221  ^◡ccnv 5665   “ cima 5669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rab 3430  df-v 3472  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-pw 4595  df-sn 4620  df-pr 4622  df-op 4626  df-br 5139  df-opab 5201  df-mpt 5222  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679

This theorem is referenced by:  isf34lem4  10351