Theorem dmxpss 5565

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

Step	Hyp	Ref	Expression
1		xpeq2 5129	. . . . . 6 |- ( B = (/) -> ( A X. B ) = ( A X. (/) ) )
2		xp0 5552	. . . . . 6 |- ( A X. (/) ) = (/)
3	1, 2	syl6eq 2672	. . . . 5 |- ( B = (/) -> ( A X. B ) = (/) )
4	3	dmeqd 5326	. . . 4 |- ( B = (/) -> dom ( A X. B ) = dom (/) )
5		dm0 5339	. . . 4 |- dom (/) = (/)
6	4, 5	syl6eq 2672	. . 3 |- ( B = (/) -> dom ( A X. B ) = (/) )
7		0ss 3972	. . 3 |- (/) C_ A
8	6, 7	syl6eqss 3655	. 2 |- ( B = (/) -> dom ( A X. B ) C_ A )
9		dmxp 5344	. . 3 |- ( B =/= (/) -> dom ( A X. B ) = A )
10		eqimss 3657	. . 3 |- ( dom ( A X. B ) = A -> dom ( A X. B ) C_ A )
11	9, 10	syl 17	. 2 |- ( B =/= (/) -> dom ( A X. B ) C_ A )
12	8, 11	pm2.61ine 2877	1 |- dom ( A X. B ) C_ A

Syntax hints:   = wceq 1483   =/= wne 2794   C_ wss 3574   (/)c0 3915   X. cxp 5112   dom cdm 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124

This theorem is referenced by:  rnxpss  5566  ssxpb  5568  funssxp  6061  dff3  6372  fparlem3  7279  fparlem4  7280  brdom3  9350  brdom5  9351  brdom4  9352  canthwelem  9472  pwfseqlem4  9484  uzrdgfni  12757  xptrrel  13719  rlimpm  14231  xpsc0  16220  xpsc1  16221  xpsfrnel2  16225  isohom  16436  ledm  17224  gsumxp  18375  dprd2d2  18443  tsmsxp  21958  dvbssntr  23664  esum2d  30155  poimirlem3  33412  rtrclex  37924  trclexi  37927  rtrclexi  37928  cnvtrcl0  37933  dmtrcl  37934  rp-imass  38065  rfovcnvf1od  38298  issmflem  40936