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Theorem dmxpss 4829
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 2618 . . . 4 𝑥 ∈ V
21eldm2 4604 . . 3 (𝑥 ∈ dom (𝐴 × 𝐵) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
3 opelxp1 4445 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥𝐴)
43exlimiv 1532 . . 3 (∃𝑦𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥𝐴)
52, 4sylbi 119 . 2 (𝑥 ∈ dom (𝐴 × 𝐵) → 𝑥𝐴)
65ssriv 3018 1 dom (𝐴 × 𝐵) ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wex 1424  wcel 1436  wss 2988  cop 3434   × cxp 4411  dom cdm 4413
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-xp 4419  df-dm 4423
This theorem is referenced by:  rnxpss  4830  dmxpss2  4831  ssxpbm  4834  ssxp1  4835  funssxp  5146  tfrlemibfn  6049  tfr1onlembfn  6065  tfrcllembfn  6078  frecuzrdgtcl  9750  frecuzrdgdomlem  9755
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