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Theorem iunpw 7123
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 3773 . . . . . . . 8 (𝑥 = 𝐴 → (𝑦𝑥𝑦 𝐴))
21biimprcd 240 . . . . . . 7 (𝑦 𝐴 → (𝑥 = 𝐴𝑦𝑥))
32reximdv 3162 . . . . . 6 (𝑦 𝐴 → (∃𝑥𝐴 𝑥 = 𝐴 → ∃𝑥𝐴 𝑦𝑥))
43com12 32 . . . . 5 (∃𝑥𝐴 𝑥 = 𝐴 → (𝑦 𝐴 → ∃𝑥𝐴 𝑦𝑥))
5 ssiun 4693 . . . . . 6 (∃𝑥𝐴 𝑦𝑥𝑦 𝑥𝐴 𝑥)
6 uniiun 4704 . . . . . 6 𝐴 = 𝑥𝐴 𝑥
75, 6syl6sseqr 3798 . . . . 5 (∃𝑥𝐴 𝑦𝑥𝑦 𝐴)
84, 7impbid1 215 . . . 4 (∃𝑥𝐴 𝑥 = 𝐴 → (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥))
9 selpw 4301 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
10 eliun 4655 . . . . 5 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥)
11 selpw 4301 . . . . . 6 (𝑦 ∈ 𝒫 𝑥𝑦𝑥)
1211rexbii 3187 . . . . 5 (∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
1310, 12bitri 264 . . . 4 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
148, 9, 133bitr4g 303 . . 3 (∃𝑥𝐴 𝑥 = 𝐴 → (𝑦 ∈ 𝒫 𝐴𝑦 𝑥𝐴 𝒫 𝑥))
1514eqrdv 2767 . 2 (∃𝑥𝐴 𝑥 = 𝐴 → 𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥)
16 ssid 3770 . . . . 5 𝐴 𝐴
17 iunpw.1 . . . . . . . 8 𝐴 ∈ V
1817uniex 7098 . . . . . . 7 𝐴 ∈ V
1918elpw 4300 . . . . . 6 ( 𝐴 ∈ 𝒫 𝐴 𝐴 𝐴)
20 eleq2 2837 . . . . . 6 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 → ( 𝐴 ∈ 𝒫 𝐴 𝐴 𝑥𝐴 𝒫 𝑥))
2119, 20syl5bbr 274 . . . . 5 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 → ( 𝐴 𝐴 𝐴 𝑥𝐴 𝒫 𝑥))
2216, 21mpbii 223 . . . 4 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 𝐴 𝑥𝐴 𝒫 𝑥)
23 eliun 4655 . . . 4 ( 𝐴 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝐴 ∈ 𝒫 𝑥)
2422, 23sylib 208 . . 3 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 → ∃𝑥𝐴 𝐴 ∈ 𝒫 𝑥)
25 elssuni 4600 . . . . . . 7 (𝑥𝐴𝑥 𝐴)
26 elpwi 4304 . . . . . . 7 ( 𝐴 ∈ 𝒫 𝑥 𝐴𝑥)
2725, 26anim12i 592 . . . . . 6 ((𝑥𝐴 𝐴 ∈ 𝒫 𝑥) → (𝑥 𝐴 𝐴𝑥))
28 eqss 3764 . . . . . 6 (𝑥 = 𝐴 ↔ (𝑥 𝐴 𝐴𝑥))
2927, 28sylibr 224 . . . . 5 ((𝑥𝐴 𝐴 ∈ 𝒫 𝑥) → 𝑥 = 𝐴)
3029ex 448 . . . 4 (𝑥𝐴 → ( 𝐴 ∈ 𝒫 𝑥𝑥 = 𝐴))
3130reximia 3155 . . 3 (∃𝑥𝐴 𝐴 ∈ 𝒫 𝑥 → ∃𝑥𝐴 𝑥 = 𝐴)
3224, 31syl 17 . 2 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 → ∃𝑥𝐴 𝑥 = 𝐴)
3315, 32impbii 199 1 (∃𝑥𝐴 𝑥 = 𝐴 ↔ 𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1629  wcel 2143  wrex 3060  Vcvv 3348  wss 3720  𝒫 cpw 4294   cuni 4571   ciun 4651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-sep 4911  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ral 3064  df-rex 3065  df-v 3350  df-in 3727  df-ss 3734  df-pw 4296  df-uni 4572  df-iun 4653
This theorem is referenced by: (None)
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