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Theorem dmrnssfld 5828
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 3483 . . . . 5 𝑥 ∈ V
21eldm2 5757 . . . 4 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
31prid1 4683 . . . . . 6 𝑥 ∈ {𝑥, 𝑦}
4 vex 3483 . . . . . . . . . 10 𝑦 ∈ V
51, 4uniop 5392 . . . . . . . . 9 𝑥, 𝑦⟩ = {𝑥, 𝑦}
61, 4uniopel 5393 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥, 𝑦⟩ ∈ 𝐴)
75, 6eqeltrrid 2921 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ∈ 𝐴)
8 elssuni 4854 . . . . . . . 8 ({𝑥, 𝑦} ∈ 𝐴 → {𝑥, 𝑦} ⊆ 𝐴)
97, 8syl 17 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ⊆ 𝐴)
109sseld 3952 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 𝐴))
113, 10mpi 20 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 𝐴)
1211exlimiv 1932 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴𝑥 𝐴)
132, 12sylbi 220 . . 3 (𝑥 ∈ dom 𝐴𝑥 𝐴)
1413ssriv 3957 . 2 dom 𝐴 𝐴
154elrn2 5808 . . . 4 (𝑦 ∈ ran 𝐴 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
164prid2 4684 . . . . . 6 𝑦 ∈ {𝑥, 𝑦}
179sseld 3952 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦 𝐴))
1816, 17mpi 20 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 𝐴)
1918exlimiv 1932 . . . 4 (∃𝑥𝑥, 𝑦⟩ ∈ 𝐴𝑦 𝐴)
2015, 19sylbi 220 . . 3 (𝑦 ∈ ran 𝐴𝑦 𝐴)
2120ssriv 3957 . 2 ran 𝐴 𝐴
2214, 21unssi 4147 1 (dom 𝐴 ∪ ran 𝐴) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wex 1781  wcel 2115  cun 3917  wss 3919  {cpr 4552  cop 4556   cuni 4824  dom cdm 5542  ran crn 5543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-cnv 5550  df-dm 5552  df-rn 5553
This theorem is referenced by:  relfld  6113  relcoi2  6115  dmexg  7608  rnexg  7609  wundm  10148  wunrn  10149  relexpdm  14402  relexprn  14406  relexpfld  14408  psdmrn  17817  dirdm  17844  dirge  17847  tailf  33783  filnetlem3  33788
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