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Theorem xpsspw 5810
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 5695 . 2 Rel (𝐴 × 𝐵)
2 opelxp 5713 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
3 snssi 4812 . . . . . . . 8 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
4 ssun3 4175 . . . . . . . 8 ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴𝐵))
53, 4syl 17 . . . . . . 7 (𝑥𝐴 → {𝑥} ⊆ (𝐴𝐵))
6 vsnex 5430 . . . . . . . 8 {𝑥} ∈ V
76elpw 4607 . . . . . . 7 ({𝑥} ∈ 𝒫 (𝐴𝐵) ↔ {𝑥} ⊆ (𝐴𝐵))
85, 7sylibr 233 . . . . . 6 (𝑥𝐴 → {𝑥} ∈ 𝒫 (𝐴𝐵))
98adantr 482 . . . . 5 ((𝑥𝐴𝑦𝐵) → {𝑥} ∈ 𝒫 (𝐴𝐵))
10 df-pr 4632 . . . . . . 7 {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
11 snssi 4812 . . . . . . . . . 10 (𝑦𝐵 → {𝑦} ⊆ 𝐵)
12 ssun4 4176 . . . . . . . . . 10 ({𝑦} ⊆ 𝐵 → {𝑦} ⊆ (𝐴𝐵))
1311, 12syl 17 . . . . . . . . 9 (𝑦𝐵 → {𝑦} ⊆ (𝐴𝐵))
145, 13anim12i 614 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → ({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)))
15 unss 4185 . . . . . . . 8 (({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)) ↔ ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1614, 15sylib 217 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1710, 16eqsstrid 4031 . . . . . 6 ((𝑥𝐴𝑦𝐵) → {𝑥, 𝑦} ⊆ (𝐴𝐵))
18 zfpair2 5429 . . . . . . 7 {𝑥, 𝑦} ∈ V
1918elpw 4607 . . . . . 6 ({𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵) ↔ {𝑥, 𝑦} ⊆ (𝐴𝐵))
2017, 19sylibr 233 . . . . 5 ((𝑥𝐴𝑦𝐵) → {𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵))
219, 20jca 513 . . . 4 ((𝑥𝐴𝑦𝐵) → ({𝑥} ∈ 𝒫 (𝐴𝐵) ∧ {𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵)))
22 prex 5433 . . . . . 6 {{𝑥}, {𝑥, 𝑦}} ∈ V
2322elpw 4607 . . . . 5 ({{𝑥}, {𝑥, 𝑦}} ∈ 𝒫 𝒫 (𝐴𝐵) ↔ {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴𝐵))
24 vex 3479 . . . . . . 7 𝑥 ∈ V
25 vex 3479 . . . . . . 7 𝑦 ∈ V
2624, 25dfop 4873 . . . . . 6 𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
2726eleq1i 2825 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴𝐵) ↔ {{𝑥}, {𝑥, 𝑦}} ∈ 𝒫 𝒫 (𝐴𝐵))
286, 18prss 4824 . . . . 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 217 . . 3 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴𝐵))
312, 30sylbi 216 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴𝐵))
321, 31relssi 5788 1 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 397  wcel 2107  cun 3947  wss 3949  𝒫 cpw 4603  {csn 4629  {cpr 4631  cop 4635   × cxp 5675
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-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-xp 5683  df-rel 5684
This theorem is referenced by:  unixpss  5811  xpexg  7737  xpfi  9317  rankxpu  9871  wunxp  10719  gruxp  10802
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