Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  opelf Structured version   Visualization version   GIF version

Theorem opelf 5978
 Description: The members of an ordered pair element of a mapping belong to the mapping's domain and codomain. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opelf ((𝐹:𝐴𝐵 ∧ ⟨𝐶, 𝐷⟩ ∈ 𝐹) → (𝐶𝐴𝐷𝐵))

Proof of Theorem opelf
StepHypRef Expression
1 fssxp 5973 . . . 4 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
21sseld 3567 . . 3 (𝐹:𝐴𝐵 → (⟨𝐶, 𝐷⟩ ∈ 𝐹 → ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)))
3 opelxp 5070 . . 3 (⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵) ↔ (𝐶𝐴𝐷𝐵))
42, 3syl6ib 240 . 2 (𝐹:𝐴𝐵 → (⟨𝐶, 𝐷⟩ ∈ 𝐹 → (𝐶𝐴𝐷𝐵)))
54imp 444 1 ((𝐹:𝐴𝐵 ∧ ⟨𝐶, 𝐷⟩ ∈ 𝐹) → (𝐶𝐴𝐷𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  ⟨cop 4131   × cxp 5036  ⟶wf 5800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808 This theorem is referenced by:  feu  5993  fcnvres  5995  fsn  6308
 Copyright terms: Public domain W3C validator