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Theorem dmrnssfld 4772
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 2663 . . . . 5 𝑥 ∈ V
21eldm2 4707 . . . 4 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
31prid1 3599 . . . . . 6 𝑥 ∈ {𝑥, 𝑦}
4 vex 2663 . . . . . . . . . 10 𝑦 ∈ V
51, 4uniop 4147 . . . . . . . . 9 𝑥, 𝑦⟩ = {𝑥, 𝑦}
61, 4uniopel 4148 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥, 𝑦⟩ ∈ 𝐴)
75, 6eqeltrrid 2205 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ∈ 𝐴)
8 elssuni 3734 . . . . . . . 8 ({𝑥, 𝑦} ∈ 𝐴 → {𝑥, 𝑦} ⊆ 𝐴)
97, 8syl 14 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ⊆ 𝐴)
109sseld 3066 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 𝐴))
113, 10mpi 15 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 𝐴)
1211exlimiv 1562 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴𝑥 𝐴)
132, 12sylbi 120 . . 3 (𝑥 ∈ dom 𝐴𝑥 𝐴)
1413ssriv 3071 . 2 dom 𝐴 𝐴
154elrn2 4751 . . . 4 (𝑦 ∈ ran 𝐴 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
164prid2 3600 . . . . . 6 𝑦 ∈ {𝑥, 𝑦}
179sseld 3066 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦 𝐴))
1816, 17mpi 15 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 𝐴)
1918exlimiv 1562 . . . 4 (∃𝑥𝑥, 𝑦⟩ ∈ 𝐴𝑦 𝐴)
2015, 19sylbi 120 . . 3 (𝑦 ∈ ran 𝐴𝑦 𝐴)
2120ssriv 3071 . 2 ran 𝐴 𝐴
2214, 21unssi 3221 1 (dom 𝐴 ∪ ran 𝐴) ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wex 1453  wcel 1465  cun 3039  wss 3041  {cpr 3498  cop 3500   cuni 3706  dom cdm 4509  ran crn 4510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-cnv 4517  df-dm 4519  df-rn 4520
This theorem is referenced by:  dmexg  4773  rnexg  4774  relfld  5037  relcoi2  5039
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