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Theorem dmrnssfld 4538
 Description: The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.)
Assertion
Ref Expression
dmrnssfld (dom A ∪ ran A) ⊆ A

Proof of Theorem dmrnssfld
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . 5 x V
21eldm2 4476 . . . 4 (x dom Ayx, y A)
31prid1 3467 . . . . . 6 x {x, y}
4 vex 2554 . . . . . . . . . 10 y V
51, 4uniop 3983 . . . . . . . . 9 x, y⟩ = {x, y}
61, 4uniopel 3984 . . . . . . . . 9 (⟨x, y Ax, y A)
75, 6syl5eqelr 2122 . . . . . . . 8 (⟨x, y A → {x, y} A)
8 elssuni 3599 . . . . . . . 8 ({x, y} A → {x, y} ⊆ A)
97, 8syl 14 . . . . . . 7 (⟨x, y A → {x, y} ⊆ A)
109sseld 2938 . . . . . 6 (⟨x, y A → (x {x, y} → x A))
113, 10mpi 15 . . . . 5 (⟨x, y Ax A)
1211exlimiv 1486 . . . 4 (yx, y Ax A)
132, 12sylbi 114 . . 3 (x dom Ax A)
1413ssriv 2943 . 2 dom A A
154elrn2 4519 . . . 4 (y ran Axx, y A)
164prid2 3468 . . . . . 6 y {x, y}
179sseld 2938 . . . . . 6 (⟨x, y A → (y {x, y} → y A))
1816, 17mpi 15 . . . . 5 (⟨x, y Ay A)
1918exlimiv 1486 . . . 4 (xx, y Ay A)
2015, 19sylbi 114 . . 3 (y ran Ay A)
2120ssriv 2943 . 2 ran A A
2214, 21unssi 3112 1 (dom A ∪ ran A) ⊆ A
 Colors of variables: wff set class Syntax hints:  ∃wex 1378   ∈ wcel 1390   ∪ cun 2909   ⊆ wss 2911  {cpr 3368  ⟨cop 3370  ∪ cuni 3571  dom cdm 4288  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-bndl 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-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-uni 3572  df-br 3756  df-opab 3810  df-cnv 4296  df-dm 4298  df-rn 4299 This theorem is referenced by:  dmexg  4539  rnexg  4540  relfld  4789  relcoi2  4791
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