MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmxpss Structured version   Unicode version

Theorem dmxpss 5348
Description: The domain of a Cartesian product is a subclass of the first factor. (Contributed by NM, 19-Mar-2007.)
Assertion
Ref Expression
dmxpss  |-  dom  ( A  X.  B )  C_  A

Proof of Theorem dmxpss
StepHypRef Expression
1 xpeq2 4928 . . . . . 6  |-  ( B  =  (/)  ->  ( A  X.  B )  =  ( A  X.  (/) ) )
2 xp0 5335 . . . . . 6  |-  ( A  X.  (/) )  =  (/)
31, 2syl6eq 2439 . . . . 5  |-  ( B  =  (/)  ->  ( A  X.  B )  =  (/) )
43dmeqd 5118 . . . 4  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  =  dom  (/) )
5 dm0 5129 . . . 4  |-  dom  (/)  =  (/)
64, 5syl6eq 2439 . . 3  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  =  (/) )
7 0ss 3741 . . 3  |-  (/)  C_  A
86, 7syl6eqss 3467 . 2  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  C_  A )
9 dmxp 5134 . . 3  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
10 eqimss 3469 . . 3  |-  ( dom  ( A  X.  B
)  =  A  ->  dom  ( A  X.  B
)  C_  A )
119, 10syl 16 . 2  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  C_  A )
128, 11pm2.61ine 2695 1  |-  dom  ( A  X.  B )  C_  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1399    =/= wne 2577    C_ wss 3389   (/)c0 3711    X. cxp 4911   dom cdm 4913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-xp 4919  df-rel 4920  df-cnv 4921  df-dm 4923
This theorem is referenced by:  rnxpss  5349  ssxpb  5351  funssxp  5652  dff3  5946  fparlem3  6801  fparlem4  6802  brdom3  8819  brdom5  8820  brdom4  8821  canthwelem  8939  pwfseqlem4  8951  uzrdgfni  11972  xptrrel  12818  rlimpm  13325  xpsc0  14967  xpsc1  14968  xpsfrnel2  14972  isohom  15182  ledm  15971  gsumxp  17118  dprd2d2  17206  tsmsxp  20742  dvbssntr  22389  esum2d  28241  rp-imass  38265
  Copyright terms: Public domain W3C validator