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Theorem iunpw 4722
 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
Assertion
Ref Expression
iunpw
Distinct variable group:   ,

Proof of Theorem iunpw
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sseq2 3334 . . . . . . . 8
21biimprcd 217 . . . . . . 7
32reximdv 2781 . . . . . 6
43com12 29 . . . . 5
5 ssiun 4097 . . . . . 6
6 uniiun 4108 . . . . . 6
75, 6syl6sseqr 3359 . . . . 5
84, 7impbid1 195 . . . 4
9 vex 2923 . . . . 5
109elpw 3769 . . . 4
11 eliun 4061 . . . . 5
12 df-pw 3765 . . . . . . 7
1312abeq2i 2515 . . . . . 6
1413rexbii 2695 . . . . 5
1511, 14bitri 241 . . . 4
168, 10, 153bitr4g 280 . . 3
1716eqrdv 2406 . 2
18 ssid 3331 . . . . 5
19 iunpw.1 . . . . . . . 8
2019uniex 4668 . . . . . . 7
2120elpw 3769 . . . . . 6
22 eleq2 2469 . . . . . 6
2321, 22syl5bbr 251 . . . . 5
2418, 23mpbii 203 . . . 4
25 eliun 4061 . . . 4
2624, 25sylib 189 . . 3
27 elssuni 4007 . . . . . . 7
28 elpwi 3771 . . . . . . 7
2927, 28anim12i 550 . . . . . 6
30 eqss 3327 . . . . . 6
3129, 30sylibr 204 . . . . 5
3231ex 424 . . . 4
3332reximia 2775 . . 3
3426, 33syl 16 . 2
3517, 34impbii 181 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wceq 1649   wcel 1721  wrex 2671  cvv 2920   wss 3284  cpw 3763  cuni 3979  ciun 4057 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-un 4664 This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-v 2922  df-in 3291  df-ss 3298  df-pw 3765  df-uni 3980  df-iun 4059
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