Theorem compss 10361

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 10356	. . 3 ⊢ ◡𝐹 = 𝐹
3	2	imaeq1i 6061	. 2 ⊢ (◡𝐹 “ 𝐺) = (𝐹 “ 𝐺)
4		difeq2 4076	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
5	4	cbvmptv 5216	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑦))
6	1, 5	eqtri 2786	. . 3 ⊢ 𝐹 = (𝑦 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑦))
7	6	mptpreima 6241	. 2 ⊢ (◡𝐹 “ 𝐺) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑦) ∈ 𝐺}
8	3, 7	eqtr3i 2788	1 ⊢ (𝐹 “ 𝐺) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑦) ∈ 𝐺}

Syntax hints:   = wceq 1570   ∈ wcel 2143  {crab 3416   ∖ cdif 3903  𝒫 cpw 4563   ↦ cmpt 5193  ^◡ccnv 5662   “ cima 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-mpt 5194  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is referenced by:  isf34lem4  10362