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Theorem rnxpss 4754
Description: The range of a cross product is a subclass of the second factor. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
rnxpss ran (𝐴 × 𝐵) ⊆ 𝐵

Proof of Theorem rnxpss
StepHypRef Expression
1 df-rn 4356 . 2 ran (𝐴 × 𝐵) = dom (𝐴 × 𝐵)
2 cnvxp 4742 . . . 4 (𝐴 × 𝐵) = (𝐵 × 𝐴)
32dmeqi 4536 . . 3 dom (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4 dmxpss 4753 . . 3 dom (𝐵 × 𝐴) ⊆ 𝐵
53, 4eqsstri 2975 . 2 dom (𝐴 × 𝐵) ⊆ 𝐵
61, 5eqsstri 2975 1 ran (𝐴 × 𝐵) ⊆ 𝐵
Colors of variables: wff set class
Syntax hints:  wss 2917   × cxp 4343  ccnv 4344  dom cdm 4345  ran crn 4346
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-eu 1903  df-mo 1904  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-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356
This theorem is referenced by:  rnxpid  4755  ssxpbm  4756  ssxp2  4758  ssrnres  4763  funssxp  5060  fconst  5082  dff2  5311  fliftf  5439
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