Theorem dmxpss 6153

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 5666	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
2		xp0 5745	. . . . . 6 ⊢ (𝐴 × ∅) = ∅
3	1, 2	eqtrdi 2812	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
4	3	dmeqd 5879	. . . 4 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = dom ∅)
5		dm0 5894	. . . 4 ⊢ dom ∅ = ∅
6	4, 5	eqtrdi 2812	. . 3 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = ∅)
7		0ss 4353	. . 3 ⊢ ∅ ⊆ 𝐴
8	6, 7	eqsstrdi 3980	. 2 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴)
9		dmxp 5903	. . 3 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
10		eqimss 3994	. . 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 3039	1 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴

Syntax hints:   = wceq 1559   ≠ wne 2956   ⊆ wss 3904  ∅c0 4285   × cxp 5643  dom cdm 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-dm 5655

This theorem is referenced by:  rnxpss  6154  ssxpb  6156  resssxp  6253  funssxp  6716  dff3  7077  fparlem3  8088  fparlem4  8089  frxp2  8119  frxp3  8126  brdom3  10482  brdom5  10483  brdom4  10484  canthwelem  10605  pwfseqlem4  10617  uzrdgfni  13968  xptrrel  14990  rlimpm  15510  isohom  17792  ledm  18605  gsumxp  19999  dprd2d2  20069  tsmsxp  24195  dvbssntr  25942  noseqrdgfn  28376  gsumpart  33204  esum2d  34351  poimirlem3  38086  rtrclex  44157  trclexi  44160  rtrclexi  44161  cnvtrcl0  44166  dmtrcl  44167  rfovcnvf1od  44544  issmflem  47265  fvconstr  49447  fvconstrn0  49448  fvconstr2  49449  fvconst0ci  49476  fvconstdomi  49477