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Theorem elreldm 4649
Description: The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.)
Assertion
Ref Expression
elreldm ((Rel 𝐴𝐵𝐴) → 𝐵 ∈ dom 𝐴)

Proof of Theorem elreldm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rel 4435 . . . . 5 (Rel 𝐴𝐴 ⊆ (V × V))
2 ssel 3017 . . . . 5 (𝐴 ⊆ (V × V) → (𝐵𝐴𝐵 ∈ (V × V)))
31, 2sylbi 119 . . . 4 (Rel 𝐴 → (𝐵𝐴𝐵 ∈ (V × V)))
4 elvv 4488 . . . 4 (𝐵 ∈ (V × V) ↔ ∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
53, 4syl6ib 159 . . 3 (Rel 𝐴 → (𝐵𝐴 → ∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩))
6 eleq1 2150 . . . . . 6 (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
7 vex 2622 . . . . . . 7 𝑥 ∈ V
8 vex 2622 . . . . . . 7 𝑦 ∈ V
97, 8opeldm 4627 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
106, 9syl6bi 161 . . . . 5 (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴𝑥 ∈ dom 𝐴))
11 inteq 3686 . . . . . . . 8 (𝐵 = ⟨𝑥, 𝑦⟩ → 𝐵 = 𝑥, 𝑦⟩)
1211inteqd 3688 . . . . . . 7 (𝐵 = ⟨𝑥, 𝑦⟩ → 𝐵 = 𝑥, 𝑦⟩)
137, 8op1stb 4290 . . . . . . 7 𝑥, 𝑦⟩ = 𝑥
1412, 13syl6eq 2136 . . . . . 6 (𝐵 = ⟨𝑥, 𝑦⟩ → 𝐵 = 𝑥)
1514eleq1d 2156 . . . . 5 (𝐵 = ⟨𝑥, 𝑦⟩ → ( 𝐵 ∈ dom 𝐴𝑥 ∈ dom 𝐴))
1610, 15sylibrd 167 . . . 4 (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴 𝐵 ∈ dom 𝐴))
1716exlimivv 1824 . . 3 (∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴 𝐵 ∈ dom 𝐴))
185, 17syli 37 . 2 (Rel 𝐴 → (𝐵𝐴 𝐵 ∈ dom 𝐴))
1918imp 122 1 ((Rel 𝐴𝐵𝐴) → 𝐵 ∈ dom 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619  wss 2997  cop 3444   cint 3683   × cxp 4426  dom cdm 4428  Rel wrel 4433
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-int 3684  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-dm 4438
This theorem is referenced by:  1stdm  5934  fundmen  6503
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