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

Theorem iunpw 7212
Description: An indexed union of a power class in terms of the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
Hypothesis
Ref Expression
iunpw.1 𝐴 ∈ V
Assertion
Ref Expression
iunpw (∃𝑥𝐴 𝑥 = 𝐴 ↔ 𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥)
Distinct variable group:   𝑥,𝐴

Proof of Theorem iunpw
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sseq2 3823 . . . . . . . 8 (𝑥 = 𝐴 → (𝑦𝑥𝑦 𝐴))
21biimprcd 242 . . . . . . 7 (𝑦 𝐴 → (𝑥 = 𝐴𝑦𝑥))
32reximdv 3196 . . . . . 6 (𝑦 𝐴 → (∃𝑥𝐴 𝑥 = 𝐴 → ∃𝑥𝐴 𝑦𝑥))
43com12 32 . . . . 5 (∃𝑥𝐴 𝑥 = 𝐴 → (𝑦 𝐴 → ∃𝑥𝐴 𝑦𝑥))
5 ssiun 4752 . . . . . 6 (∃𝑥𝐴 𝑦𝑥𝑦 𝑥𝐴 𝑥)
6 uniiun 4763 . . . . . 6 𝐴 = 𝑥𝐴 𝑥
75, 6syl6sseqr 3848 . . . . 5 (∃𝑥𝐴 𝑦𝑥𝑦 𝐴)
84, 7impbid1 217 . . . 4 (∃𝑥𝐴 𝑥 = 𝐴 → (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥))
9 selpw 4356 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
10 eliun 4714 . . . . 5 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥)
11 selpw 4356 . . . . . 6 (𝑦 ∈ 𝒫 𝑥𝑦𝑥)
1211rexbii 3222 . . . . 5 (∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
1310, 12bitri 267 . . . 4 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
148, 9, 133bitr4g 306 . . 3 (∃𝑥𝐴 𝑥 = 𝐴 → (𝑦 ∈ 𝒫 𝐴𝑦 𝑥𝐴 𝒫 𝑥))
1514eqrdv 2797 . 2 (∃𝑥𝐴 𝑥 = 𝐴 → 𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥)
16 ssid 3819 . . . . 5 𝐴 𝐴
17 iunpw.1 . . . . . . . 8 𝐴 ∈ V
1817uniex 7187 . . . . . . 7 𝐴 ∈ V
1918elpw 4355 . . . . . 6 ( 𝐴 ∈ 𝒫 𝐴 𝐴 𝐴)
20 eleq2 2867 . . . . . 6 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 → ( 𝐴 ∈ 𝒫 𝐴 𝐴 𝑥𝐴 𝒫 𝑥))
2119, 20syl5bbr 277 . . . . 5 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 → ( 𝐴 𝐴 𝐴 𝑥𝐴 𝒫 𝑥))
2216, 21mpbii 225 . . . 4 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 𝐴 𝑥𝐴 𝒫 𝑥)
23 eliun 4714 . . . 4 ( 𝐴 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝐴 ∈ 𝒫 𝑥)
2422, 23sylib 210 . . 3 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 → ∃𝑥𝐴 𝐴 ∈ 𝒫 𝑥)
25 elssuni 4659 . . . . . . 7 (𝑥𝐴𝑥 𝐴)
26 elpwi 4359 . . . . . . 7 ( 𝐴 ∈ 𝒫 𝑥 𝐴𝑥)
2725, 26anim12i 607 . . . . . 6 ((𝑥𝐴 𝐴 ∈ 𝒫 𝑥) → (𝑥 𝐴 𝐴𝑥))
28 eqss 3813 . . . . . 6 (𝑥 = 𝐴 ↔ (𝑥 𝐴 𝐴𝑥))
2927, 28sylibr 226 . . . . 5 ((𝑥𝐴 𝐴 ∈ 𝒫 𝑥) → 𝑥 = 𝐴)
3029ex 402 . . . 4 (𝑥𝐴 → ( 𝐴 ∈ 𝒫 𝑥𝑥 = 𝐴))
3130reximia 3189 . . 3 (∃𝑥𝐴 𝐴 ∈ 𝒫 𝑥 → ∃𝑥𝐴 𝑥 = 𝐴)
3224, 31syl 17 . 2 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 → ∃𝑥𝐴 𝑥 = 𝐴)
3315, 32impbii 201 1 (∃𝑥𝐴 𝑥 = 𝐴 ↔ 𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385   = wceq 1653  wcel 2157  wrex 3090  Vcvv 3385  wss 3769  𝒫 cpw 4349   cuni 4628   ciun 4710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777  ax-sep 4975  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-v 3387  df-in 3776  df-ss 3783  df-pw 4351  df-uni 4629  df-iun 4712
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator