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Theorem iunpw 4161
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 2944 . . . . . . . 8 (x = A → (yxy A))
21biimprcd 149 . . . . . . 7 (y A → (x = Ayx))
32reximdv 2398 . . . . . 6 (y A → (x A x = Ax A yx))
43com12 27 . . . . 5 (x A x = A → (y Ax A yx))
5 ssiun 3673 . . . . . 6 (x A yxy x A x)
6 uniiun 3684 . . . . . 6 A = x A x
75, 6syl6sseqr 2969 . . . . 5 (x A yxy A)
84, 7impbid1 130 . . . 4 (x A x = A → (y Ax A yx))
9 vex 2538 . . . . 5 y V
109elpw 3340 . . . 4 (y 𝒫 Ay A)
11 eliun 3635 . . . . 5 (y x A 𝒫 xx A y 𝒫 x)
12 df-pw 3336 . . . . . . 7 𝒫 x = {yyx}
1312abeq2i 2130 . . . . . 6 (y 𝒫 xyx)
1413rexbii 2309 . . . . 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 2020 . 2 (x A x = A → 𝒫 A = x A 𝒫 x)
18 ssid 2941 . . . . 5 A A
19 iunpw.1 . . . . . . . 8 A V
2019uniex 4124 . . . . . . 7 A V
2120elpw 3340 . . . . . 6 ( A 𝒫 A A A)
22 eleq2 2083 . . . . . 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 3635 . . . 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 3582 . . . . . . 7 (x Ax A)
28 elpwi 3343 . . . . . . 7 ( A 𝒫 x Ax)
2927, 28anim12i 321 . . . . . 6 ((x A A 𝒫 x) → (x A Ax))
30 eqss 2937 . . . . . 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 2392 . . 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 1228   wcel 1374  wrex 2285  Vcvv 2535  wss 2894  𝒫 cpw 3334   cuni 3554   ciun 3631
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-in 2901  df-ss 2908  df-pw 3336  df-uni 3555  df-iun 3633
This theorem is referenced by: (None)
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