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Theorem iunpw 4751
 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 3362 . . . . . . . 8
21biimprcd 217 . . . . . . 7
32reximdv 2809 . . . . . 6
43com12 29 . . . . 5
5 ssiun 4125 . . . . . 6
6 uniiun 4136 . . . . . 6
75, 6syl6sseqr 3387 . . . . 5
84, 7impbid1 195 . . . 4
9 vex 2951 . . . . 5
109elpw 3797 . . . 4
11 eliun 4089 . . . . 5
12 df-pw 3793 . . . . . . 7
1312abeq2i 2542 . . . . . 6
1413rexbii 2722 . . . . 5
1511, 14bitri 241 . . . 4
168, 10, 153bitr4g 280 . . 3
1716eqrdv 2433 . 2
18 ssid 3359 . . . . 5
19 iunpw.1 . . . . . . . 8
2019uniex 4697 . . . . . . 7
2120elpw 3797 . . . . . 6
22 eleq2 2496 . . . . . 6
2321, 22syl5bbr 251 . . . . 5
2418, 23mpbii 203 . . . 4
25 eliun 4089 . . . 4
2624, 25sylib 189 . . 3
27 elssuni 4035 . . . . . . 7
28 elpwi 3799 . . . . . . 7
2927, 28anim12i 550 . . . . . 6
30 eqss 3355 . . . . . 6
3129, 30sylibr 204 . . . . 5
3231ex 424 . . . 4
3332reximia 2803 . . 3
3426, 33syl 16 . 2
3517, 34impbii 181 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wceq 1652   wcel 1725  wrex 2698  cvv 2948   wss 3312  cpw 3791  cuni 4007  ciun 4085 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-in 3319  df-ss 3326  df-pw 3793  df-uni 4008  df-iun 4087
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