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Theorem pwtrVD 41148
Description: Virtual deduction proof of pwtr 5336; see pwtrrVD 41149 for the converse. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
pwtrVD (Tr 𝐴 → Tr 𝒫 𝐴)

Proof of Theorem pwtrVD
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 5165 . . 3 (Tr 𝒫 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴))
2 idn1 40898 . . . . . . 7 (   Tr 𝐴   ▶   Tr 𝐴   )
3 idn2 40937 . . . . . . . . . 10 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   )
4 simpr 487 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑦 ∈ 𝒫 𝐴)
53, 4e2 40955 . . . . . . . . 9 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑦 ∈ 𝒫 𝐴   )
6 elpwi 4549 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
75, 6e2 40955 . . . . . . . 8 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑦𝐴   )
8 simpl 485 . . . . . . . . 9 ((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧𝑦)
93, 8e2 40955 . . . . . . . 8 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑧𝑦   )
10 ssel 3959 . . . . . . . 8 (𝑦𝐴 → (𝑧𝑦𝑧𝐴))
117, 9, 10e22 40995 . . . . . . 7 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑧𝐴   )
12 trss 5172 . . . . . . 7 (Tr 𝐴 → (𝑧𝐴𝑧𝐴))
132, 11, 12e12 41048 . . . . . 6 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑧𝐴   )
14 vex 3496 . . . . . . 7 𝑧 ∈ V
1514elpw 4544 . . . . . 6 (𝑧 ∈ 𝒫 𝐴𝑧𝐴)
1613, 15e2bir 40957 . . . . 5 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑧 ∈ 𝒫 𝐴   )
1716in2 40929 . . . 4 (   Tr 𝐴   ▶   ((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴)   )
1817gen12 40942 . . 3 (   Tr 𝐴   ▶   𝑧𝑦((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴)   )
19 biimpr 222 . . 3 ((Tr 𝒫 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴)) → (∀𝑧𝑦((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴) → Tr 𝒫 𝐴))
201, 18, 19e01 41015 . 2 (   Tr 𝐴   ▶   Tr 𝒫 𝐴   )
2120in1 40895 1 (Tr 𝐴 → Tr 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1528  wcel 2107  wss 3934  𝒫 cpw 4537  Tr wtr 5163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-v 3495  df-in 3941  df-ss 3950  df-pw 4539  df-uni 4831  df-tr 5164  df-vd1 40894  df-vd2 40902
This theorem is referenced by: (None)
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