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Theorem compsscnvlem 10407
Description: Lemma for compsscnv 10408. (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 4145 . . . 4 (𝐴𝑥) ⊆ 𝐴
31, 2eqsstrdi 4049 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑦𝐴)
4 velpw 4609 . . 3 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
53, 4sylibr 234 . 2 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑦 ∈ 𝒫 𝐴)
61difeq2d 4135 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝐴𝑦) = (𝐴 ∖ (𝐴𝑥)))
7 elpwi 4611 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
87adantr 480 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → 𝑥𝐴)
9 dfss4 4274 . . . 4 (𝑥𝐴 ↔ (𝐴 ∖ (𝐴𝑥)) = 𝑥)
108, 9sylib 218 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝐴 ∖ (𝐴𝑥)) = 𝑥)
116, 10eqtr2d 2775 . 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 1536  wcel 2105  cdif 3959  wss 3962  𝒫 cpw 4604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-in 3969  df-ss 3979  df-pw 4606
This theorem is referenced by:  compsscnv  10408
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