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Theorem rnxpss2 4864
 Description: Upper bound for the range of a binary relation. (Contributed by BJ, 10-Jul-2022.)
Assertion
Ref Expression
rnxpss2 (𝑅 ⊆ (𝐴 × 𝐵) → ran 𝑅𝐵)

Proof of Theorem rnxpss2
StepHypRef Expression
1 rnss 4665 . 2 (𝑅 ⊆ (𝐴 × 𝐵) → ran 𝑅 ⊆ ran (𝐴 × 𝐵))
2 rnxpss 4862 . 2 ran (𝐴 × 𝐵) ⊆ 𝐵
31, 2syl6ss 3037 1 (𝑅 ⊆ (𝐴 × 𝐵) → ran 𝑅𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ⊆ wss 2999   × cxp 4436  ran crn 4439 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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036 This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449 This theorem is referenced by:  cossxp2  4954
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