Theorem dmxpss 6160

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 5672	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
2		xp0 5751	. . . . . 6 ⊢ (𝐴 × ∅) = ∅
3	1, 2	eqtrdi 2816	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
4	3	dmeqd 5885	. . . 4 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = dom ∅)
5		dm0 5900	. . . 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 5909	. . 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 5649  dom cdm 5651

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 5250  ax-pr 5394

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 5105  df-opab 5167  df-xp 5657  df-dm 5661

This theorem is referenced by:  rnxpss  6161  ssxpb  6163  resssxp  6260  funssxp  6724  dff3  7085  fparlem3  8097  fparlem4  8098  frxp2  8128  frxp3  8135  brdom3  10500  brdom5  10501  brdom4  10502  canthwelem  10623  pwfseqlem4  10635  uzrdgfni  13982  xptrrel  15005  rlimpm  15539  isohom  17821  ledm  18634  gsumxp  20034  dprd2d2  20104  tsmsxp  24269  dvbssntr  26016  noseqrdgfn  28453  gsumpart  33291  esum2d  34395  poimirlem3  38129  rtrclex  44200  trclexi  44203  rtrclexi  44204  cnvtrcl0  44209  dmtrcl  44210  rfovcnvf1od  44587  issmflem  47300  fvconstr  49492  fvconstrn0  49493  fvconstr2  49494  fvconst0ci  49521  fvconstdomi  49522