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Theorem dmxpss2 4939
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 4706 . 2 (𝑅 ⊆ (𝐴 × 𝐵) → dom 𝑅 ⊆ dom (𝐴 × 𝐵))
2 dmxpss 4937 . 2 dom (𝐴 × 𝐵) ⊆ 𝐴
31, 2syl6ss 3077 1 (𝑅 ⊆ (𝐴 × 𝐵) → dom 𝑅𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wss 3039   × cxp 4505  dom cdm 4507
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-dm 4517
This theorem is referenced by:  cossxp2  5030
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