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Theorem rniun 4677
 Description: The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
rniun ran x A B = x A ran B

Proof of Theorem rniun
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 2571 . . . 4 (x A yy, z Byx Ay, z B)
2 vex 2554 . . . . . 6 z V
32elrn2 4519 . . . . 5 (z ran Byy, z B)
43rexbii 2325 . . . 4 (x A z ran Bx A yy, z B)
5 eliun 3652 . . . . 5 (⟨y, z x A Bx Ay, z B)
65exbii 1493 . . . 4 (yy, z x A Byx Ay, z B)
71, 4, 63bitr4ri 202 . . 3 (yy, z x A Bx A z ran B)
82elrn2 4519 . . 3 (z ran x A Byy, z x A B)
9 eliun 3652 . . 3 (z x A ran Bx A z ran B)
107, 8, 93bitr4i 201 . 2 (z ran x A Bz x A ran B)
1110eqriv 2034 1 ran x A B = x A ran B
 Colors of variables: wff set class Syntax hints:   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301  ⟨cop 3370  ∪ ciun 3648  ran crn 4289 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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  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-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iun 3650  df-br 3756  df-opab 3810  df-cnv 4296  df-dm 4298  df-rn 4299 This theorem is referenced by:  rnuni  4678  fun11iun  5090
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