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Theorem dmrnssfld 5416
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 3234 . . . . 5 𝑥 ∈ V
21eldm2 5354 . . . 4 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
31prid1 4329 . . . . . 6 𝑥 ∈ {𝑥, 𝑦}
4 vex 3234 . . . . . . . . . 10 𝑦 ∈ V
51, 4uniop 5006 . . . . . . . . 9 𝑥, 𝑦⟩ = {𝑥, 𝑦}
61, 4uniopel 5007 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥, 𝑦⟩ ∈ 𝐴)
75, 6syl5eqelr 2735 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ∈ 𝐴)
8 elssuni 4499 . . . . . . . 8 ({𝑥, 𝑦} ∈ 𝐴 → {𝑥, 𝑦} ⊆ 𝐴)
97, 8syl 17 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ⊆ 𝐴)
109sseld 3635 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 𝐴))
113, 10mpi 20 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 𝐴)
1211exlimiv 1898 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴𝑥 𝐴)
132, 12sylbi 207 . . 3 (𝑥 ∈ dom 𝐴𝑥 𝐴)
1413ssriv 3640 . 2 dom 𝐴 𝐴
154elrn2 5397 . . . 4 (𝑦 ∈ ran 𝐴 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
164prid2 4330 . . . . . 6 𝑦 ∈ {𝑥, 𝑦}
179sseld 3635 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦 𝐴))
1816, 17mpi 20 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 𝐴)
1918exlimiv 1898 . . . 4 (∃𝑥𝑥, 𝑦⟩ ∈ 𝐴𝑦 𝐴)
2015, 19sylbi 207 . . 3 (𝑦 ∈ ran 𝐴𝑦 𝐴)
2120ssriv 3640 . 2 ran 𝐴 𝐴
2214, 21unssi 3821 1 (dom 𝐴 ∪ ran 𝐴) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wex 1744  wcel 2030  cun 3605  wss 3607  {cpr 4212  cop 4216   cuni 4468  dom cdm 5143  ran crn 5144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-cnv 5151  df-dm 5153  df-rn 5154
This theorem is referenced by:  relfld  5699  relcoi2  5701  dmexg  7139  rnexg  7140  wundm  9588  wunrn  9589  relexpdm  13827  relexprn  13831  relexpfld  13833  psdmrn  17254  dirdm  17281  dirge  17284  tailf  32495  filnetlem3  32500
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