ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmxpss GIF version

Theorem dmxpss 4804
Description: The domain of a cross product is a subclass of the first factor. (Contributed by NM, 19-Mar-2007.)
Assertion
Ref Expression
dmxpss dom (𝐴 × 𝐵) ⊆ 𝐴

Proof of Theorem dmxpss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2613 . . . 4 𝑥 ∈ V
21eldm2 4582 . . 3 (𝑥 ∈ dom (𝐴 × 𝐵) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
3 opelxp1 4424 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥𝐴)
43exlimiv 1530 . . 3 (∃𝑦𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥𝐴)
52, 4sylbi 119 . 2 (𝑥 ∈ dom (𝐴 × 𝐵) → 𝑥𝐴)
65ssriv 3013 1 dom (𝐴 × 𝐵) ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wex 1422  wcel 1434  wss 2983  cop 3420   × cxp 4390  dom cdm 4392
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-xp 4398  df-dm 4402
This theorem is referenced by:  rnxpss  4805  ssxpbm  4807  ssxp1  4808  funssxp  5112  tfrlemibfn  5998  tfr1onlembfn  6014  tfrcllembfn  6027  frecuzrdgtcl  9530  frecuzrdgdomlem  9535
  Copyright terms: Public domain W3C validator