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Theorem rniun 4661
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 2554 . . . 4 (x A yy, z Byx Ay, z B)
2 vex 2538 . . . . . 6 z V
32elrn2 4503 . . . . 5 (z ran Byy, z B)
43rexbii 2309 . . . 4 (x A z ran Bx A yy, z B)
5 eliun 3635 . . . . 5 (⟨y, z x A Bx Ay, z B)
65exbii 1478 . . . 4 (yy, z x A Byx Ay, z B)
71, 4, 63bitr4ri 202 . . 3 (yy, z x A Bx A z ran B)
82elrn2 4503 . . 3 (z ran x A Byy, z x A B)
9 eliun 3635 . . 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 2019 1 ran x A B = x A ran B
Colors of variables: wff set class
Syntax hints:   = wceq 1228  wex 1362   wcel 1374  wrex 2285  cop 3353   ciun 3631  ran crn 4273
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-iun 3633  df-br 3739  df-opab 3793  df-cnv 4280  df-dm 4282  df-rn 4283
This theorem is referenced by:  rnuni  4662  fun11iun  5072
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