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Theorem elreldm 5687
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 5450 . . . . 5 (Rel 𝐴𝐴 ⊆ (V × V))
2 ssel 3883 . . . . 5 (𝐴 ⊆ (V × V) → (𝐵𝐴𝐵 ∈ (V × V)))
31, 2sylbi 218 . . . 4 (Rel 𝐴 → (𝐵𝐴𝐵 ∈ (V × V)))
4 elvv 5512 . . . 4 (𝐵 ∈ (V × V) ↔ ∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
53, 4syl6ib 252 . . 3 (Rel 𝐴 → (𝐵𝐴 → ∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩))
6 eleq1 2870 . . . . . 6 (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
7 vex 3440 . . . . . . 7 𝑥 ∈ V
8 vex 3440 . . . . . . 7 𝑦 ∈ V
97, 8opeldm 5662 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
106, 9syl6bi 254 . . . . 5 (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴𝑥 ∈ dom 𝐴))
11 inteq 4785 . . . . . . . 8 (𝐵 = ⟨𝑥, 𝑦⟩ → 𝐵 = 𝑥, 𝑦⟩)
1211inteqd 4787 . . . . . . 7 (𝐵 = ⟨𝑥, 𝑦⟩ → 𝐵 = 𝑥, 𝑦⟩)
137, 8op1stb 5255 . . . . . . 7 𝑥, 𝑦⟩ = 𝑥
1412, 13syl6eq 2847 . . . . . 6 (𝐵 = ⟨𝑥, 𝑦⟩ → 𝐵 = 𝑥)
1514eleq1d 2867 . . . . 5 (𝐵 = ⟨𝑥, 𝑦⟩ → ( 𝐵 ∈ dom 𝐴𝑥 ∈ dom 𝐴))
1610, 15sylibrd 260 . . . 4 (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴 𝐵 ∈ dom 𝐴))
1716exlimivv 1910 . . 3 (∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩ → (𝐵𝐴 𝐵 ∈ dom 𝐴))
185, 17syli 39 . 2 (Rel 𝐴 → (𝐵𝐴 𝐵 ∈ dom 𝐴))
1918imp 407 1 ((Rel 𝐴𝐵𝐴) → 𝐵 ∈ dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wex 1761  wcel 2081  Vcvv 3437  wss 3859  cop 4478   cint 4782   × cxp 5441  dom cdm 5443  Rel wrel 5448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-int 4783  df-br 4963  df-opab 5025  df-xp 5449  df-rel 5450  df-dm 5453
This theorem is referenced by:  1stdm  7595  fundmen  8431
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