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Theorem dmrnssfld 5987
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 3482 . . . . 5 𝑥 ∈ V
21eldm2 5915 . . . 4 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
31prid1 4767 . . . . . 6 𝑥 ∈ {𝑥, 𝑦}
4 vex 3482 . . . . . . . . . 10 𝑦 ∈ V
51, 4uniop 5525 . . . . . . . . 9 𝑥, 𝑦⟩ = {𝑥, 𝑦}
61, 4uniopel 5526 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥, 𝑦⟩ ∈ 𝐴)
75, 6eqeltrrid 2844 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ∈ 𝐴)
8 elssuni 4942 . . . . . . . 8 ({𝑥, 𝑦} ∈ 𝐴 → {𝑥, 𝑦} ⊆ 𝐴)
97, 8syl 17 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ⊆ 𝐴)
109sseld 3994 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 𝐴))
113, 10mpi 20 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 𝐴)
1211exlimiv 1928 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴𝑥 𝐴)
132, 12sylbi 217 . . 3 (𝑥 ∈ dom 𝐴𝑥 𝐴)
1413ssriv 3999 . 2 dom 𝐴 𝐴
154elrn2 5906 . . . 4 (𝑦 ∈ ran 𝐴 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
164prid2 4768 . . . . . 6 𝑦 ∈ {𝑥, 𝑦}
179sseld 3994 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦 𝐴))
1816, 17mpi 20 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 𝐴)
1918exlimiv 1928 . . . 4 (∃𝑥𝑥, 𝑦⟩ ∈ 𝐴𝑦 𝐴)
2015, 19sylbi 217 . . 3 (𝑦 ∈ ran 𝐴𝑦 𝐴)
2120ssriv 3999 . 2 ran 𝐴 𝐴
2214, 21unssi 4201 1 (dom 𝐴 ∪ ran 𝐴) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wex 1776  wcel 2106  cun 3961  wss 3963  {cpr 4633  cop 4637   cuni 4912  dom cdm 5689  ran crn 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700
This theorem is referenced by:  relfld  6297  relcoi2  6299  dmexg  7924  rnexg  7925  wundm  10766  wunrn  10767  relexpdm  15079  relexprn  15083  relexpfld  15085  psdmrn  18631  dirdm  18658  dirge  18661  tailf  36358  filnetlem3  36363  dmwf  44940  rnwf  44941
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