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Theorem xpsspw 5680
 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 5571 . 2 Rel (𝐴 × 𝐵)
2 opelxp 5589 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
3 snssi 4739 . . . . . . . 8 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
4 ssun3 4153 . . . . . . . 8 ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴𝐵))
53, 4syl 17 . . . . . . 7 (𝑥𝐴 → {𝑥} ⊆ (𝐴𝐵))
6 snex 5327 . . . . . . . 8 {𝑥} ∈ V
76elpw 4548 . . . . . . 7 ({𝑥} ∈ 𝒫 (𝐴𝐵) ↔ {𝑥} ⊆ (𝐴𝐵))
85, 7sylibr 235 . . . . . 6 (𝑥𝐴 → {𝑥} ∈ 𝒫 (𝐴𝐵))
98adantr 481 . . . . 5 ((𝑥𝐴𝑦𝐵) → {𝑥} ∈ 𝒫 (𝐴𝐵))
10 df-pr 4566 . . . . . . 7 {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
11 snssi 4739 . . . . . . . . . 10 (𝑦𝐵 → {𝑦} ⊆ 𝐵)
12 ssun4 4154 . . . . . . . . . 10 ({𝑦} ⊆ 𝐵 → {𝑦} ⊆ (𝐴𝐵))
1311, 12syl 17 . . . . . . . . 9 (𝑦𝐵 → {𝑦} ⊆ (𝐴𝐵))
145, 13anim12i 612 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → ({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)))
15 unss 4163 . . . . . . . 8 (({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)) ↔ ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1614, 15sylib 219 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1710, 16eqsstrid 4018 . . . . . 6 ((𝑥𝐴𝑦𝐵) → {𝑥, 𝑦} ⊆ (𝐴𝐵))
18 zfpair2 5326 . . . . . . 7 {𝑥, 𝑦} ∈ V
1918elpw 4548 . . . . . 6 ({𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵) ↔ {𝑥, 𝑦} ⊆ (𝐴𝐵))
2017, 19sylibr 235 . . . . 5 ((𝑥𝐴𝑦𝐵) → {𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵))
219, 20jca 512 . . . 4 ((𝑥𝐴𝑦𝐵) → ({𝑥} ∈ 𝒫 (𝐴𝐵) ∧ {𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵)))
22 prex 5328 . . . . . 6 {{𝑥}, {𝑥, 𝑦}} ∈ V
2322elpw 4548 . . . . 5 ({{𝑥}, {𝑥, 𝑦}} ∈ 𝒫 𝒫 (𝐴𝐵) ↔ {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴𝐵))
24 vex 3502 . . . . . . 7 𝑥 ∈ V
25 vex 3502 . . . . . . 7 𝑦 ∈ V
2624, 25dfop 4800 . . . . . 6 𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
2726eleq1i 2907 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴𝐵) ↔ {{𝑥}, {𝑥, 𝑦}} ∈ 𝒫 𝒫 (𝐴𝐵))
286, 18prss 4751 . . . . 5 (({𝑥} ∈ 𝒫 (𝐴𝐵) ∧ {𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵)) ↔ {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴𝐵))
2923, 27, 283bitr4ri 305 . . . 4 (({𝑥} ∈ 𝒫 (𝐴𝐵) ∧ {𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴𝐵))
3021, 29sylib 219 . . 3 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴𝐵))
312, 30sylbi 218 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴𝐵))
321, 31relssi 5658 1 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 396   ∈ wcel 2106   ∪ cun 3937   ⊆ wss 3939  𝒫 cpw 4541  {csn 4563  {cpr 4565  ⟨cop 4569   × cxp 5551 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 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-opab 5125  df-xp 5559  df-rel 5560 This theorem is referenced by:  unixpss  5681  xpexg  7465  rankxpu  9297  wunxp  10138  gruxp  10221
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