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Theorem dmxpss 5054
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 2740 . . . 4 𝑥 ∈ V
21eldm2 4820 . . 3 (𝑥 ∈ dom (𝐴 × 𝐵) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
3 opelxp1 4656 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥𝐴)
43exlimiv 1598 . . 3 (∃𝑦𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥𝐴)
52, 4sylbi 121 . 2 (𝑥 ∈ dom (𝐴 × 𝐵) → 𝑥𝐴)
65ssriv 3159 1 dom (𝐴 × 𝐵) ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wex 1492  wcel 2148  wss 3129  cop 3594   × cxp 4620  dom cdm 4622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4628  df-dm 4632
This theorem is referenced by:  rnxpss  5055  dmxpss2  5056  ssxpbm  5059  ssxp1  5060  funssxp  5380  tfrlemibfn  6322  tfr1onlembfn  6338  tfrcllembfn  6351  frecuzrdgtcl  10385  frecuzrdgdomlem  10390  dvbssntrcntop  13786
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