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Theorem elpglem3 47244
Description: Lemma for elpg 47245. (Contributed by Emmett Weisz, 28-Aug-2021.)
Assertion
Ref Expression
elpglem3 (∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem elpglem3
StepHypRef Expression
1 vex 3448 . . . . . . . 8 𝑥 ∈ V
2 pweq 4575 . . . . . . . . . 10 (𝑦 = 𝑥 → 𝒫 𝑦 = 𝒫 𝑥)
32sqxpeqd 5666 . . . . . . . . 9 (𝑦 = 𝑥 → (𝒫 𝑦 × 𝒫 𝑦) = (𝒫 𝑥 × 𝒫 𝑥))
4 eqid 2733 . . . . . . . . 9 (𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦)) = (𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))
51pwex 5336 . . . . . . . . . 10 𝒫 𝑥 ∈ V
65, 5xpex 7688 . . . . . . . . 9 (𝒫 𝑥 × 𝒫 𝑥) ∈ V
73, 4, 6fvmpt 6949 . . . . . . . 8 (𝑥 ∈ V → ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥) = (𝒫 𝑥 × 𝒫 𝑥))
81, 7ax-mp 5 . . . . . . 7 ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥) = (𝒫 𝑥 × 𝒫 𝑥)
98eleq2i 2826 . . . . . 6 (𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥) ↔ 𝐴 ∈ (𝒫 𝑥 × 𝒫 𝑥))
10 elxp7 7957 . . . . . 6 (𝐴 ∈ (𝒫 𝑥 × 𝒫 𝑥) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)))
119, 10bitri 275 . . . . 5 (𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)))
1211anbi2i 624 . . . 4 ((𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ (𝑥 ⊆ Pg ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))))
13 an12 644 . . . 4 ((𝑥 ⊆ Pg ∧ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))) ↔ (𝐴 ∈ (V × V) ∧ (𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))))
1412, 13bitri 275 . . 3 ((𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ (𝐴 ∈ (V × V) ∧ (𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))))
1514exbii 1851 . 2 (∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ ∃𝑥(𝐴 ∈ (V × V) ∧ (𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))))
16 19.42v 1958 . 2 (∃𝑥(𝐴 ∈ (V × V) ∧ (𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))) ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))))
1715, 16bitri 275 1 (∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  Vcvv 3444  wss 3911  𝒫 cpw 4561  cmpt 5189   × cxp 5632  cfv 6497  1st c1st 7920  2nd c2nd 7921  Pgcpg 47240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-1st 7922  df-2nd 7923
This theorem is referenced by:  elpg  47245
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