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Theorem elreldm 4853
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 4633 . . . . 5 (Rel 𝐴𝐴 ⊆ (V × V))
2 ssel 3149 . . . . 5 (𝐴 ⊆ (V × V) → (𝐵𝐴𝐵 ∈ (V × V)))
31, 2sylbi 121 . . . 4 (Rel 𝐴 → (𝐵𝐴𝐵 ∈ (V × V)))
4 elvv 4688 . . . 4 (𝐵 ∈ (V × V) ↔ ∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
53, 4imbitrdi 161 . . 3 (Rel 𝐴 → (𝐵𝐴 → ∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩))
6 eleq1 2240 . . . . . 6 (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
7 vex 2740 . . . . . . 7 𝑥 ∈ V
8 vex 2740 . . . . . . 7 𝑦 ∈ V
97, 8opeldm 4830 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
106, 9syl6bi 163 . . . . 5 (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴𝑥 ∈ dom 𝐴))
11 inteq 3847 . . . . . . . 8 (𝐵 = ⟨𝑥, 𝑦⟩ → 𝐵 = 𝑥, 𝑦⟩)
1211inteqd 3849 . . . . . . 7 (𝐵 = ⟨𝑥, 𝑦⟩ → 𝐵 = 𝑥, 𝑦⟩)
137, 8op1stb 4478 . . . . . . 7 𝑥, 𝑦⟩ = 𝑥
1412, 13eqtrdi 2226 . . . . . 6 (𝐵 = ⟨𝑥, 𝑦⟩ → 𝐵 = 𝑥)
1514eleq1d 2246 . . . . 5 (𝐵 = ⟨𝑥, 𝑦⟩ → ( 𝐵 ∈ dom 𝐴𝑥 ∈ dom 𝐴))
1610, 15sylibrd 169 . . . 4 (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴 𝐵 ∈ dom 𝐴))
1716exlimivv 1896 . . 3 (∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴 𝐵 ∈ dom 𝐴))
185, 17syli 37 . 2 (Rel 𝐴 → (𝐵𝐴 𝐵 ∈ dom 𝐴))
1918imp 124 1 ((Rel 𝐴𝐵𝐴) → 𝐵 ∈ dom 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wex 1492  wcel 2148  Vcvv 2737  wss 3129  cop 3595   cint 3844   × cxp 4624  dom cdm 4626  Rel wrel 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-int 3845  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-dm 4636
This theorem is referenced by:  1stdm  6182  fundmen  6805
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