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Theorem dmxpss 4976
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 2692 . . . 4 𝑥 ∈ V
21eldm2 4744 . . 3 (𝑥 ∈ dom (𝐴 × 𝐵) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
3 opelxp1 4580 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥𝐴)
43exlimiv 1578 . . 3 (∃𝑦𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥𝐴)
52, 4sylbi 120 . 2 (𝑥 ∈ dom (𝐴 × 𝐵) → 𝑥𝐴)
65ssriv 3105 1 dom (𝐴 × 𝐵) ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wex 1469  wcel 1481  wss 3075  cop 3534   × cxp 4544  dom cdm 4546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-xp 4552  df-dm 4556
This theorem is referenced by:  rnxpss  4977  dmxpss2  4978  ssxpbm  4981  ssxp1  4982  funssxp  5299  tfrlemibfn  6232  tfr1onlembfn  6248  tfrcllembfn  6261  frecuzrdgtcl  10215  frecuzrdgdomlem  10220  dvbssntrcntop  12859
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