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Theorem dmrnssfld 5121
 Description: The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.)
Assertion
Ref Expression
dmrnssfld

Proof of Theorem dmrnssfld
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2951 . . . . 5
21eldm2 5060 . . . 4
31prid1 3904 . . . . . 6
4 vex 2951 . . . . . . . . . 10
51, 4uniop 4451 . . . . . . . . 9
61, 4uniopel 4452 . . . . . . . . 9
75, 6syl5eqelr 2520 . . . . . . . 8
8 elssuni 4035 . . . . . . . 8
97, 8syl 16 . . . . . . 7
109sseld 3339 . . . . . 6
113, 10mpi 17 . . . . 5
1211exlimiv 1644 . . . 4
132, 12sylbi 188 . . 3
1413ssriv 3344 . 2
154elrn2 5101 . . . 4
164prid2 3905 . . . . . 6
179sseld 3339 . . . . . 6
1816, 17mpi 17 . . . . 5
1918exlimiv 1644 . . . 4
2015, 19sylbi 188 . . 3
2120ssriv 3344 . 2
2214, 21unssi 3514 1
 Colors of variables: wff set class Syntax hints:  wex 1550   wcel 1725   cun 3310   wss 3312  cpr 3807  cop 3809  cuni 4007   cdm 4870   crn 4871 This theorem is referenced by:  dmexg  5122  rnexg  5123  relfld  5387  relcoi2  5389  wundm  8595  wunrn  8596  psdmrn  14631  dirdm  14671  dirge  14674  tailf  26395  filnetlem3  26400 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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-cnv 4878  df-dm 4880  df-rn 4881
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