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Theorem dmxpss2 4826
Description: Upper bound for the domain of a binary relation. (Contributed by BJ, 10-Jul-2022.)
Assertion
Ref Expression
dmxpss2 (𝑅 ⊆ (𝐴 × 𝐵) → dom 𝑅𝐴)

Proof of Theorem dmxpss2
StepHypRef Expression
1 dmss 4602 . 2 (𝑅 ⊆ (𝐴 × 𝐵) → dom 𝑅 ⊆ dom (𝐴 × 𝐵))
2 dmxpss 4824 . 2 dom (𝐴 × 𝐵) ⊆ 𝐴
31, 2syl6ss 3026 1 (𝑅 ⊆ (𝐴 × 𝐵) → dom 𝑅𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wss 2988   × cxp 4408  dom cdm 4410
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 3931  ax-pow 3983  ax-pr 4009
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 3821  df-opab 3875  df-xp 4416  df-dm 4420
This theorem is referenced by:  cossxp2  4917
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