Theorem compss 10399

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 10394	. . 3 ⊢ ◡𝐹 = 𝐹
3	2	imaeq1i 6057	. 2 ⊢ (◡𝐹 “ 𝐺) = (𝐹 “ 𝐺)
4		difeq2 4102	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
5	4	cbvmptv 5237	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑦))
6	1, 5	eqtri 2757	. . 3 ⊢ 𝐹 = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑦))
7	6	mptpreima 6240	. 2 ⊢ (◡𝐹 “ 𝐺) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑦) ∈ 𝐺}
8	3, 7	eqtr3i 2759	1 ⊢ (𝐹 “ 𝐺) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑦) ∈ 𝐺}

Syntax hints:   = wceq 1539   ∈ wcel 2107  {crab 3420   ∖ cdif 3930  𝒫 cpw 4582   ↦ cmpt 5207  ^◡ccnv 5666   “ cima 5670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-br 5126  df-opab 5188  df-mpt 5208  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680

This theorem is referenced by:  isf34lem4  10400