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Theorem iunpw 4177
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 (x A x = A ↔ 𝒫 A = x A 𝒫 x)
Distinct variable group:   x,A

Proof of Theorem iunpw
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 sseq2 2961 . . . . . . . 8 (x = A → (yxy A))
21biimprcd 149 . . . . . . 7 (y A → (x = Ayx))
32reximdv 2414 . . . . . 6 (y A → (x A x = Ax A yx))
43com12 27 . . . . 5 (x A x = A → (y Ax A yx))
5 ssiun 3690 . . . . . 6 (x A yxy x A x)
6 uniiun 3701 . . . . . 6 A = x A x
75, 6syl6sseqr 2986 . . . . 5 (x A yxy A)
84, 7impbid1 130 . . . 4 (x A x = A → (y Ax A yx))
9 vex 2554 . . . . 5 y V
109elpw 3357 . . . 4 (y 𝒫 Ay A)
11 eliun 3652 . . . . 5 (y x A 𝒫 xx A y 𝒫 x)
12 df-pw 3353 . . . . . . 7 𝒫 x = {yyx}
1312abeq2i 2145 . . . . . 6 (y 𝒫 xyx)
1413rexbii 2325 . . . . 5 (x A y 𝒫 xx A yx)
1511, 14bitri 173 . . . 4 (y x A 𝒫 xx A yx)
168, 10, 153bitr4g 212 . . 3 (x A x = A → (y 𝒫 Ay x A 𝒫 x))
1716eqrdv 2035 . 2 (x A x = A → 𝒫 A = x A 𝒫 x)
18 ssid 2958 . . . . 5 A A
19 iunpw.1 . . . . . . . 8 A V
2019uniex 4140 . . . . . . 7 A V
2120elpw 3357 . . . . . 6 ( A 𝒫 A A A)
22 eleq2 2098 . . . . . 6 (𝒫 A = x A 𝒫 x → ( A 𝒫 A A x A 𝒫 x))
2321, 22syl5bbr 183 . . . . 5 (𝒫 A = x A 𝒫 x → ( A A A x A 𝒫 x))
2418, 23mpbii 136 . . . 4 (𝒫 A = x A 𝒫 x A x A 𝒫 x)
25 eliun 3652 . . . 4 ( A x A 𝒫 xx A A 𝒫 x)
2624, 25sylib 127 . . 3 (𝒫 A = x A 𝒫 xx A A 𝒫 x)
27 elssuni 3599 . . . . . . 7 (x Ax A)
28 elpwi 3360 . . . . . . 7 ( A 𝒫 x Ax)
2927, 28anim12i 321 . . . . . 6 ((x A A 𝒫 x) → (x A Ax))
30 eqss 2954 . . . . . 6 (x = A ↔ (x A Ax))
3129, 30sylibr 137 . . . . 5 ((x A A 𝒫 x) → x = A)
3231ex 108 . . . 4 (x A → ( A 𝒫 xx = A))
3332reximia 2408 . . 3 (x A A 𝒫 xx A x = A)
3426, 33syl 14 . 2 (𝒫 A = x A 𝒫 xx A x = A)
3517, 34impbii 117 1 (x A x = A ↔ 𝒫 A = x A 𝒫 x)
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242   wcel 1390  wrex 2301  Vcvv 2551  wss 2911  𝒫 cpw 3351   cuni 3571   ciun 3648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-uni 3572  df-iun 3650
This theorem is referenced by: (None)
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