MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpsspw Structured version   Visualization version   GIF version

Theorem xpsspw 5401
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 5295 . 2 Rel (𝐴 × 𝐵)
2 opelxp 5313 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
3 snssi 4493 . . . . . . . 8 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
4 ssun3 3940 . . . . . . . 8 ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴𝐵))
53, 4syl 17 . . . . . . 7 (𝑥𝐴 → {𝑥} ⊆ (𝐴𝐵))
6 snex 5064 . . . . . . . 8 {𝑥} ∈ V
76elpw 4321 . . . . . . 7 ({𝑥} ∈ 𝒫 (𝐴𝐵) ↔ {𝑥} ⊆ (𝐴𝐵))
85, 7sylibr 225 . . . . . 6 (𝑥𝐴 → {𝑥} ∈ 𝒫 (𝐴𝐵))
98adantr 472 . . . . 5 ((𝑥𝐴𝑦𝐵) → {𝑥} ∈ 𝒫 (𝐴𝐵))
10 df-pr 4337 . . . . . . 7 {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
11 snssi 4493 . . . . . . . . . 10 (𝑦𝐵 → {𝑦} ⊆ 𝐵)
12 ssun4 3941 . . . . . . . . . 10 ({𝑦} ⊆ 𝐵 → {𝑦} ⊆ (𝐴𝐵))
1311, 12syl 17 . . . . . . . . 9 (𝑦𝐵 → {𝑦} ⊆ (𝐴𝐵))
145, 13anim12i 606 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → ({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)))
15 unss 3949 . . . . . . . 8 (({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)) ↔ ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1614, 15sylib 209 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1710, 16syl5eqss 3809 . . . . . 6 ((𝑥𝐴𝑦𝐵) → {𝑥, 𝑦} ⊆ (𝐴𝐵))
18 zfpair2 5063 . . . . . . 7 {𝑥, 𝑦} ∈ V
1918elpw 4321 . . . . . 6 ({𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵) ↔ {𝑥, 𝑦} ⊆ (𝐴𝐵))
2017, 19sylibr 225 . . . . 5 ((𝑥𝐴𝑦𝐵) → {𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵))
219, 20jca 507 . . . 4 ((𝑥𝐴𝑦𝐵) → ({𝑥} ∈ 𝒫 (𝐴𝐵) ∧ {𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵)))
22 prex 5065 . . . . . 6 {{𝑥}, {𝑥, 𝑦}} ∈ V
2322elpw 4321 . . . . 5 ({{𝑥}, {𝑥, 𝑦}} ∈ 𝒫 𝒫 (𝐴𝐵) ↔ {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴𝐵))
24 vex 3353 . . . . . . 7 𝑥 ∈ V
25 vex 3353 . . . . . . 7 𝑦 ∈ V
2624, 25dfop 4558 . . . . . 6 𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
2726eleq1i 2835 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴𝐵) ↔ {{𝑥}, {𝑥, 𝑦}} ∈ 𝒫 𝒫 (𝐴𝐵))
286, 18prss 4505 . . . . 5 (({𝑥} ∈ 𝒫 (𝐴𝐵) ∧ {𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵)) ↔ {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴𝐵))
2923, 27, 283bitr4ri 295 . . . 4 (({𝑥} ∈ 𝒫 (𝐴𝐵) ∧ {𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴𝐵))
3021, 29sylib 209 . . 3 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴𝐵))
312, 30sylbi 208 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴𝐵))
321, 31relssi 5380 1 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 384  wcel 2155  cun 3730  wss 3732  𝒫 cpw 4315  {csn 4334  {cpr 4336  cop 4340   × cxp 5275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-opab 4872  df-xp 5283  df-rel 5284
This theorem is referenced by:  unixpss  5402  xpexg  7158  rankxpu  8954  wunxp  9799  gruxp  9882
  Copyright terms: Public domain W3C validator