Theorem dmxpss 6123

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 5640	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
2		xp0 5719	. . . . . 6 ⊢ (𝐴 × ∅) = ∅
3	1, 2	eqtrdi 2784	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
4	3	dmeqd 5849	. . . 4 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = dom ∅)
5		dm0 5864	. . . 4 ⊢ dom ∅ = ∅
6	4, 5	eqtrdi 2784	. . 3 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = ∅)
7		0ss 4349	. . 3 ⊢ ∅ ⊆ 𝐴
8	6, 7	eqsstrdi 3975	. 2 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴)
9		dmxp 5873	. . 3 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
10		eqimss 3989	. . 3 ⊢ (dom (𝐴 × 𝐵) = 𝐴 → dom (𝐴 × 𝐵) ⊆ 𝐴)
11	9, 10	syl 17	. 2 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴)
12	8, 11	pm2.61ine 3012	1 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴

Syntax hints:   = wceq 1541   ≠ wne 2929   ⊆ wss 3898  ∅c0 4282   × cxp 5617  dom cdm 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-xp 5625  df-dm 5629

This theorem is referenced by:  rnxpss  6124  ssxpb  6126  resssxp  6222  funssxp  6684  dff3  7039  fparlem3  8050  fparlem4  8051  frxp2  8080  frxp3  8087  brdom3  10426  brdom5  10427  brdom4  10428  canthwelem  10548  pwfseqlem4  10560  uzrdgfni  13867  xptrrel  14889  rlimpm  15409  isohom  17685  ledm  18498  gsumxp  19890  dprd2d2  19960  tsmsxp  24071  dvbssntr  25829  noseqrdgfn  28237  gsumpart  33044  esum2d  34127  poimirlem3  37683  rtrclex  43734  trclexi  43737  rtrclexi  43738  cnvtrcl0  43743  dmtrcl  43744  rfovcnvf1od  44121  issmflem  46849  fvconstr  48986  fvconstrn0  48987  fvconstr2  48988  fvconst0ci  49015  fvconstdomi  49016