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Theorem xpsspw 5822
Description: A Cartesian product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.)
Assertion
Ref Expression
xpsspw (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)

Proof of Theorem xpsspw
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 5707 . 2 Rel (𝐴 × 𝐵)
2 opelxp 5725 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
3 snssi 4813 . . . . . . . 8 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
4 ssun3 4190 . . . . . . . 8 ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴𝐵))
53, 4syl 17 . . . . . . 7 (𝑥𝐴 → {𝑥} ⊆ (𝐴𝐵))
6 vsnex 5440 . . . . . . . 8 {𝑥} ∈ V
76elpw 4609 . . . . . . 7 ({𝑥} ∈ 𝒫 (𝐴𝐵) ↔ {𝑥} ⊆ (𝐴𝐵))
85, 7sylibr 234 . . . . . 6 (𝑥𝐴 → {𝑥} ∈ 𝒫 (𝐴𝐵))
98adantr 480 . . . . 5 ((𝑥𝐴𝑦𝐵) → {𝑥} ∈ 𝒫 (𝐴𝐵))
10 df-pr 4634 . . . . . . 7 {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
11 snssi 4813 . . . . . . . . . 10 (𝑦𝐵 → {𝑦} ⊆ 𝐵)
12 ssun4 4191 . . . . . . . . . 10 ({𝑦} ⊆ 𝐵 → {𝑦} ⊆ (𝐴𝐵))
1311, 12syl 17 . . . . . . . . 9 (𝑦𝐵 → {𝑦} ⊆ (𝐴𝐵))
145, 13anim12i 613 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → ({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)))
15 unss 4200 . . . . . . . 8 (({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)) ↔ ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1614, 15sylib 218 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1710, 16eqsstrid 4044 . . . . . 6 ((𝑥𝐴𝑦𝐵) → {𝑥, 𝑦} ⊆ (𝐴𝐵))
18 zfpair2 5439 . . . . . . 7 {𝑥, 𝑦} ∈ V
1918elpw 4609 . . . . . 6 ({𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵) ↔ {𝑥, 𝑦} ⊆ (𝐴𝐵))
2017, 19sylibr 234 . . . . 5 ((𝑥𝐴𝑦𝐵) → {𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵))
219, 20jca 511 . . . 4 ((𝑥𝐴𝑦𝐵) → ({𝑥} ∈ 𝒫 (𝐴𝐵) ∧ {𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵)))
22 prex 5443 . . . . . 6 {{𝑥}, {𝑥, 𝑦}} ∈ V
2322elpw 4609 . . . . 5 ({{𝑥}, {𝑥, 𝑦}} ∈ 𝒫 𝒫 (𝐴𝐵) ↔ {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴𝐵))
24 vex 3482 . . . . . . 7 𝑥 ∈ V
25 vex 3482 . . . . . . 7 𝑦 ∈ V
2624, 25dfop 4877 . . . . . 6 𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
2726eleq1i 2830 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴𝐵) ↔ {{𝑥}, {𝑥, 𝑦}} ∈ 𝒫 𝒫 (𝐴𝐵))
286, 18prss 4825 . . . . 5 (({𝑥} ∈ 𝒫 (𝐴𝐵) ∧ {𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵)) ↔ {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴𝐵))
2923, 27, 283bitr4ri 304 . . . 4 (({𝑥} ∈ 𝒫 (𝐴𝐵) ∧ {𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴𝐵))
3021, 29sylib 218 . . 3 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴𝐵))
312, 30sylbi 217 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴𝐵))
321, 31relssi 5800 1 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 395  wcel 2106  cun 3961  wss 3963  𝒫 cpw 4605  {csn 4631  {cpr 4633  cop 4637   × cxp 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695  df-rel 5696
This theorem is referenced by:  unixpss  5823  xpexg  7769  xpfi  9356  rankxpu  9914  wunxp  10762  gruxp  10845  xpwf  44939
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