Theorem dmxpss 6161

Description: The domain of a Cartesian product is included in its first factor. (Contributed by NM, 19-Mar-2007.)

Assertion

Ref	Expression
dmxpss	⊢ dom (𝐴 × 𝐵) ⊆ 𝐴

Proof of Theorem dmxpss

Step	Hyp	Ref	Expression
1		xpeq2 5673	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
2		xp0 5752	. . . . . 6 ⊢ (𝐴 × ∅) = ∅
3	1, 2	eqtrdi 2816	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
4	3	dmeqd 5886	. . . 4 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = dom ∅)
5		dm0 5901	. . . 4 ⊢ dom ∅ = ∅
6	4, 5	eqtrdi 2816	. . 3 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = ∅)
7		0ss 4357	. . 3 ⊢ ∅ ⊆ 𝐴
8	6, 7	eqsstrdi 3983	. 2 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴)
9		dmxp 5910	. . 3 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
10		eqimss 3997	. . 3 ⊢ (dom (𝐴 × 𝐵) = 𝐴 → dom (𝐴 × 𝐵) ⊆ 𝐴)
11	9, 10	syl 18	. 2 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴)
12	8, 11	pm2.61ine 3043	1 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴

Syntax hints:   = wceq 1563   ≠ wne 2960   ⊆ wss 3907  ∅c0 4288   × cxp 5650  dom cdm 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-dm 5662

This theorem is referenced by:  rnxpss  6162  ssxpb  6164  resssxp  6261  funssxp  6724  dff3  7085  fparlem3  8097  fparlem4  8098  frxp2  8128  frxp3  8135  brdom3  10500  brdom5  10501  brdom4  10502  canthwelem  10623  pwfseqlem4  10635  uzrdgfni  13985  xptrrel  15007  rlimpm  15541  isohom  17823  ledm  18636  gsumxp  20037  dprd2d2  20107  tsmsxp  24273  dvbssntr  26020  noseqrdgfn  28457  gsumpart  33296  esum2d  34400  poimirlem3  38134  rtrclex  44205  trclexi  44208  rtrclexi  44209  cnvtrcl0  44214  dmtrcl  44215  rfovcnvf1od  44592  issmflem  47299  fvconstr  49491  fvconstrn0  49492  fvconstr2  49493  fvconst0ci  49520  fvconstdomi  49521