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Theorem iunpw 7791
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 4010 . . . . . . . 8 (𝑥 = 𝐴 → (𝑦𝑥𝑦 𝐴))
21biimprcd 250 . . . . . . 7 (𝑦 𝐴 → (𝑥 = 𝐴𝑦𝑥))
32reximdv 3170 . . . . . 6 (𝑦 𝐴 → (∃𝑥𝐴 𝑥 = 𝐴 → ∃𝑥𝐴 𝑦𝑥))
43com12 32 . . . . 5 (∃𝑥𝐴 𝑥 = 𝐴 → (𝑦 𝐴 → ∃𝑥𝐴 𝑦𝑥))
5 ssiun 5046 . . . . . 6 (∃𝑥𝐴 𝑦𝑥𝑦 𝑥𝐴 𝑥)
6 uniiun 5058 . . . . . 6 𝐴 = 𝑥𝐴 𝑥
75, 6sseqtrrdi 4025 . . . . 5 (∃𝑥𝐴 𝑦𝑥𝑦 𝐴)
84, 7impbid1 225 . . . 4 (∃𝑥𝐴 𝑥 = 𝐴 → (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥))
9 velpw 4605 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
10 eliun 4995 . . . . 5 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥)
11 velpw 4605 . . . . . 6 (𝑦 ∈ 𝒫 𝑥𝑦𝑥)
1211rexbii 3094 . . . . 5 (∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
1310, 12bitri 275 . . . 4 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
148, 9, 133bitr4g 314 . . 3 (∃𝑥𝐴 𝑥 = 𝐴 → (𝑦 ∈ 𝒫 𝐴𝑦 𝑥𝐴 𝒫 𝑥))
1514eqrdv 2735 . 2 (∃𝑥𝐴 𝑥 = 𝐴 → 𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥)
16 ssid 4006 . . . . 5 𝐴 𝐴
17 iunpw.1 . . . . . . . 8 𝐴 ∈ V
1817uniex 7761 . . . . . . 7 𝐴 ∈ V
1918elpw 4604 . . . . . 6 ( 𝐴 ∈ 𝒫 𝐴 𝐴 𝐴)
20 eleq2 2830 . . . . . 6 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 → ( 𝐴 ∈ 𝒫 𝐴 𝐴 𝑥𝐴 𝒫 𝑥))
2119, 20bitr3id 285 . . . . 5 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 → ( 𝐴 𝐴 𝐴 𝑥𝐴 𝒫 𝑥))
2216, 21mpbii 233 . . . 4 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 𝐴 𝑥𝐴 𝒫 𝑥)
23 eliun 4995 . . . 4 ( 𝐴 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝐴 ∈ 𝒫 𝑥)
2422, 23sylib 218 . . 3 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 → ∃𝑥𝐴 𝐴 ∈ 𝒫 𝑥)
25 elssuni 4937 . . . . . . 7 (𝑥𝐴𝑥 𝐴)
26 elpwi 4607 . . . . . . 7 ( 𝐴 ∈ 𝒫 𝑥 𝐴𝑥)
2725, 26anim12i 613 . . . . . 6 ((𝑥𝐴 𝐴 ∈ 𝒫 𝑥) → (𝑥 𝐴 𝐴𝑥))
28 eqss 3999 . . . . . 6 (𝑥 = 𝐴 ↔ (𝑥 𝐴 𝐴𝑥))
2927, 28sylibr 234 . . . . 5 ((𝑥𝐴 𝐴 ∈ 𝒫 𝑥) → 𝑥 = 𝐴)
3029ex 412 . . . 4 (𝑥𝐴 → ( 𝐴 ∈ 𝒫 𝑥𝑥 = 𝐴))
3130reximia 3081 . . 3 (∃𝑥𝐴 𝐴 ∈ 𝒫 𝑥 → ∃𝑥𝐴 𝑥 = 𝐴)
3224, 31syl 17 . 2 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 → ∃𝑥𝐴 𝑥 = 𝐴)
3315, 32impbii 209 1 (∃𝑥𝐴 𝑥 = 𝐴 ↔ 𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  wss 3951  𝒫 cpw 4600   cuni 4907   ciun 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-ss 3968  df-pw 4602  df-uni 4908  df-iun 4993
This theorem is referenced by: (None)
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