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Theorem dmxpss 4753
 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 2560 . . . 4 𝑥 ∈ V
21eldm2 4533 . . 3 (𝑥 ∈ dom (𝐴 × 𝐵) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
3 opelxp1 4377 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥𝐴)
43exlimiv 1489 . . 3 (∃𝑦𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥𝐴)
52, 4sylbi 114 . 2 (𝑥 ∈ dom (𝐴 × 𝐵) → 𝑥𝐴)
65ssriv 2949 1 dom (𝐴 × 𝐵) ⊆ 𝐴
 Colors of variables: wff set class Syntax hints:  ∃wex 1381   ∈ wcel 1393   ⊆ wss 2917  ⟨cop 3378   × cxp 4343  dom cdm 4345 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-dm 4355 This theorem is referenced by:  rnxpss  4754  ssxpbm  4756  ssxp1  4757  funssxp  5060  tfrlemibfn  5942  frecuzrdgfn  9198
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