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Theorem dmuni 4488
Description: The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 3-Feb-2004.)
Assertion
Ref Expression
dmuni  dom  U.  U_  dom
Distinct variable group:   ,

Proof of Theorem dmuni
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 excom 1551 . . . . 5  <. ,  >.  <. , 
>.
2 ancom 253 . . . . . . 7  <. , 
>.  <. , 
>.
3 19.41v 1779 . . . . . . 7  <. , 
>.  <. , 
>.
4 vex 2554 . . . . . . . . 9 
_V
54eldm2 4476 . . . . . . . 8  dom  <. ,  >.
65anbi2i 430 . . . . . . 7  dom  <. ,  >.
72, 3, 63bitr4i 201 . . . . . 6  <. , 
>.  dom
87exbii 1493 . . . . 5  <. ,  >.  dom
91, 8bitri 173 . . . 4  <. ,  >.  dom
10 eluni 3574 . . . . 5  <. ,  >.  U.  <. ,  >.
1110exbii 1493 . . . 4  <. , 
>.  U.  <. ,  >.
12 df-rex 2306 . . . 4  dom  dom
139, 11, 123bitr4i 201 . . 3  <. , 
>.  U.  dom
144eldm2 4476 . . 3  dom  U.  <. , 
>.  U.
15 eliun 3652 . . 3  U_  dom  dom
1613, 14, 153bitr4i 201 . 2  dom  U.  U_  dom
1716eqriv 2034 1  dom  U.  U_  dom
Colors of variables: wff set class
Syntax hints:   wa 97   wceq 1242  wex 1378   wcel 1390  wrex 2301   <.cop 3370   U.cuni 3571   U_ciun 3648   dom cdm 4288
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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  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-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-dm 4298
This theorem is referenced by:  tfrlem8  5875  tfrlemi14d  5888
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