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Theorem dmiun 4487
Description: The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)
Assertion
Ref Expression
dmiun dom x A B = x A dom B

Proof of Theorem dmiun
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 2571 . . . 4 (x A zy, z Bzx Ay, z B)
2 vex 2554 . . . . . 6 y V
32eldm2 4476 . . . . 5 (y dom Bzy, z B)
43rexbii 2325 . . . 4 (x A y dom Bx A zy, z B)
5 eliun 3652 . . . . 5 (⟨y, z x A Bx Ay, z B)
65exbii 1493 . . . 4 (zy, z x A Bzx Ay, z B)
71, 4, 63bitr4ri 202 . . 3 (zy, z x A Bx A y dom B)
82eldm2 4476 . . 3 (y dom x A Bzy, z x A B)
9 eliun 3652 . . 3 (y x A dom Bx A y dom B)
107, 8, 93bitr4i 201 . 2 (y dom x A By x A dom B)
1110eqriv 2034 1 dom x A B = x A dom B
Colors of variables: wff set class
Syntax hints:   = wceq 1242  wex 1378   wcel 1390  wrex 2301  cop 3370   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-iun 3650  df-br 3756  df-dm 4298
This theorem is referenced by: (None)
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