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Theorem iunpw 4369
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 3089 . . . . . . . 8 (𝑥 = 𝐴 → (𝑦𝑥𝑦 𝐴))
21biimprcd 159 . . . . . . 7 (𝑦 𝐴 → (𝑥 = 𝐴𝑦𝑥))
32reximdv 2508 . . . . . 6 (𝑦 𝐴 → (∃𝑥𝐴 𝑥 = 𝐴 → ∃𝑥𝐴 𝑦𝑥))
43com12 30 . . . . 5 (∃𝑥𝐴 𝑥 = 𝐴 → (𝑦 𝐴 → ∃𝑥𝐴 𝑦𝑥))
5 ssiun 3823 . . . . . 6 (∃𝑥𝐴 𝑦𝑥𝑦 𝑥𝐴 𝑥)
6 uniiun 3834 . . . . . 6 𝐴 = 𝑥𝐴 𝑥
75, 6sseqtrrdi 3114 . . . . 5 (∃𝑥𝐴 𝑦𝑥𝑦 𝐴)
84, 7impbid1 141 . . . 4 (∃𝑥𝐴 𝑥 = 𝐴 → (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥))
9 vex 2661 . . . . 5 𝑦 ∈ V
109elpw 3484 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
11 eliun 3785 . . . . 5 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥)
12 df-pw 3480 . . . . . . 7 𝒫 𝑥 = {𝑦𝑦𝑥}
1312abeq2i 2226 . . . . . 6 (𝑦 ∈ 𝒫 𝑥𝑦𝑥)
1413rexbii 2417 . . . . 5 (∃𝑥𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
1511, 14bitri 183 . . . 4 (𝑦 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
168, 10, 153bitr4g 222 . . 3 (∃𝑥𝐴 𝑥 = 𝐴 → (𝑦 ∈ 𝒫 𝐴𝑦 𝑥𝐴 𝒫 𝑥))
1716eqrdv 2113 . 2 (∃𝑥𝐴 𝑥 = 𝐴 → 𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥)
18 ssid 3085 . . . . 5 𝐴 𝐴
19 iunpw.1 . . . . . . . 8 𝐴 ∈ V
2019uniex 4327 . . . . . . 7 𝐴 ∈ V
2120elpw 3484 . . . . . 6 ( 𝐴 ∈ 𝒫 𝐴 𝐴 𝐴)
22 eleq2 2179 . . . . . 6 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 → ( 𝐴 ∈ 𝒫 𝐴 𝐴 𝑥𝐴 𝒫 𝑥))
2321, 22syl5bbr 193 . . . . 5 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 → ( 𝐴 𝐴 𝐴 𝑥𝐴 𝒫 𝑥))
2418, 23mpbii 147 . . . 4 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 𝐴 𝑥𝐴 𝒫 𝑥)
25 eliun 3785 . . . 4 ( 𝐴 𝑥𝐴 𝒫 𝑥 ↔ ∃𝑥𝐴 𝐴 ∈ 𝒫 𝑥)
2624, 25sylib 121 . . 3 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 → ∃𝑥𝐴 𝐴 ∈ 𝒫 𝑥)
27 elssuni 3732 . . . . . . 7 (𝑥𝐴𝑥 𝐴)
28 elpwi 3487 . . . . . . 7 ( 𝐴 ∈ 𝒫 𝑥 𝐴𝑥)
2927, 28anim12i 334 . . . . . 6 ((𝑥𝐴 𝐴 ∈ 𝒫 𝑥) → (𝑥 𝐴 𝐴𝑥))
30 eqss 3080 . . . . . 6 (𝑥 = 𝐴 ↔ (𝑥 𝐴 𝐴𝑥))
3129, 30sylibr 133 . . . . 5 ((𝑥𝐴 𝐴 ∈ 𝒫 𝑥) → 𝑥 = 𝐴)
3231ex 114 . . . 4 (𝑥𝐴 → ( 𝐴 ∈ 𝒫 𝑥𝑥 = 𝐴))
3332reximia 2502 . . 3 (∃𝑥𝐴 𝐴 ∈ 𝒫 𝑥 → ∃𝑥𝐴 𝑥 = 𝐴)
3426, 33syl 14 . 2 (𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥 → ∃𝑥𝐴 𝑥 = 𝐴)
3517, 34impbii 125 1 (∃𝑥𝐴 𝑥 = 𝐴 ↔ 𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥)
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1314  wcel 1463  wrex 2392  Vcvv 2658  wss 3039  𝒫 cpw 3478   cuni 3704   ciun 3781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-in 3045  df-ss 3052  df-pw 3480  df-uni 3705  df-iun 3783
This theorem is referenced by: (None)
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