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Theorem dmxpss 5271
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 4852 . . . . . 6  |-  ( B  =  (/)  ->  ( A  X.  B )  =  ( A  X.  (/) ) )
2 xp0 5258 . . . . . 6  |-  ( A  X.  (/) )  =  (/)
31, 2syl6eq 2503 . . . . 5  |-  ( B  =  (/)  ->  ( A  X.  B )  =  (/) )
43dmeqd 5040 . . . 4  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  =  dom  (/) )
5 dm0 5051 . . . 4  |-  dom  (/)  =  (/)
64, 5syl6eq 2503 . . 3  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  =  (/) )
7 0ss 3765 . . 3  |-  (/)  C_  A
86, 7syl6eqss 3484 . 2  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  C_  A )
9 dmxp 5056 . . 3  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
10 eqimss 3486 . . 3  |-  ( dom  ( A  X.  B
)  =  A  ->  dom  ( A  X.  B
)  C_  A )
119, 10syl 17 . 2  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  C_  A )
128, 11pm2.61ine 2709 1  |-  dom  ( A  X.  B )  C_  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1446    =/= wne 2624    C_ wss 3406   (/)c0 3733    X. cxp 4835   dom cdm 4837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-br 4406  df-opab 4465  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847
This theorem is referenced by:  rnxpss  5272  ssxpb  5274  funssxp  5747  dff3  6040  fparlem3  6903  fparlem4  6904  brdom3  8961  brdom5  8962  brdom4  8963  canthwelem  9080  pwfseqlem4  9092  uzrdgfni  12179  xptrrel  13056  rlimpm  13576  xpsc0  15478  xpsc1  15479  xpsfrnel2  15483  isohom  15693  ledm  16482  gsumxp  17620  dprd2d2  17689  tsmsxp  21181  dvbssntr  22867  esum2d  28926  poimirlem3  31955  rtrclex  36236  trclexi  36239  rtrclexi  36240  cnvtrcl0  36245  dmtrcl  36246  rp-imass  36378
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