MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  compsscnvlem Structured version   Visualization version   GIF version

Theorem compsscnvlem 10410
Description: Lemma for compsscnv 10411. (Contributed by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
compsscnvlem ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)))
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem compsscnvlem
StepHypRef Expression
1 simpr 484 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑦 = (𝐴𝑥))
2 difss 4136 . . . 4 (𝐴𝑥) ⊆ 𝐴
31, 2eqsstrdi 4028 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑦𝐴)
4 velpw 4605 . . 3 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
53, 4sylibr 234 . 2 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑦 ∈ 𝒫 𝐴)
61difeq2d 4126 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝐴𝑦) = (𝐴 ∖ (𝐴𝑥)))
7 elpwi 4607 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
87adantr 480 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑥𝐴)
9 dfss4 4269 . . . 4 (𝑥𝐴 ↔ (𝐴 ∖ (𝐴𝑥)) = 𝑥)
108, 9sylib 218 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝐴 ∖ (𝐴𝑥)) = 𝑥)
116, 10eqtr2d 2778 . 2 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑥 = (𝐴𝑦))
125, 11jca 511 1 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cdif 3948  wss 3951  𝒫 cpw 4600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968  df-pw 4602
This theorem is referenced by:  compsscnv  10411
  Copyright terms: Public domain W3C validator