ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmxpss2 GIF version

Theorem dmxpss2 5053
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 4819 . 2 (𝑅 ⊆ (𝐴 × 𝐵) → dom 𝑅 ⊆ dom (𝐴 × 𝐵))
2 dmxpss 5051 . 2 dom (𝐴 × 𝐵) ⊆ 𝐴
31, 2sstrdi 3165 1 (𝑅 ⊆ (𝐴 × 𝐵) → dom 𝑅𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wss 3127   × cxp 4618  dom cdm 4620
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-dm 4630
This theorem is referenced by:  cossxp2  5144
  Copyright terms: Public domain W3C validator