ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmrnssfld GIF version

Theorem dmrnssfld 4652
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 2615 . . . . 5 𝑥 ∈ V
21eldm2 4590 . . . 4 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
31prid1 3522 . . . . . 6 𝑥 ∈ {𝑥, 𝑦}
4 vex 2615 . . . . . . . . . 10 𝑦 ∈ V
51, 4uniop 4045 . . . . . . . . 9 𝑥, 𝑦⟩ = {𝑥, 𝑦}
61, 4uniopel 4046 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥, 𝑦⟩ ∈ 𝐴)
75, 6syl5eqelr 2170 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ∈ 𝐴)
8 elssuni 3655 . . . . . . . 8 ({𝑥, 𝑦} ∈ 𝐴 → {𝑥, 𝑦} ⊆ 𝐴)
97, 8syl 14 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ⊆ 𝐴)
109sseld 3009 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 𝐴))
113, 10mpi 15 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 𝐴)
1211exlimiv 1530 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴𝑥 𝐴)
132, 12sylbi 119 . . 3 (𝑥 ∈ dom 𝐴𝑥 𝐴)
1413ssriv 3014 . 2 dom 𝐴 𝐴
154elrn2 4633 . . . 4 (𝑦 ∈ ran 𝐴 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
164prid2 3523 . . . . . 6 𝑦 ∈ {𝑥, 𝑦}
179sseld 3009 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦 𝐴))
1816, 17mpi 15 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 𝐴)
1918exlimiv 1530 . . . 4 (∃𝑥𝑥, 𝑦⟩ ∈ 𝐴𝑦 𝐴)
2015, 19sylbi 119 . . 3 (𝑦 ∈ ran 𝐴𝑦 𝐴)
2120ssriv 3014 . 2 ran 𝐴 𝐴
2214, 21unssi 3159 1 (dom 𝐴 ∪ ran 𝐴) ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wex 1422  wcel 1434  cun 2982  wss 2984  {cpr 3423  cop 3425   cuni 3627  dom cdm 4399  ran crn 4400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-cnv 4407  df-dm 4409  df-rn 4410
This theorem is referenced by:  dmexg  4653  rnexg  4654  relfld  4911  relcoi2  4913
  Copyright terms: Public domain W3C validator