Theorem dmxpm 4903

Description: The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
dmxpm	⊢ (∃𝑥 𝑥 ∈ 𝐵 → dom (𝐴 × 𝐵) = 𝐴)

Distinct variable group:   𝑥,𝐵

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem dmxpm

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2269	. . 3 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
2	1	cbvexv 1943	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵)
3		df-xp 4685	. . . 4 ⊢ (𝐴 × 𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}
4	3	dmeqi 4884	. . 3 ⊢ dom (𝐴 × 𝐵) = dom {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}
5		id 19	. . . . 5 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → ∃𝑧 𝑧 ∈ 𝐵)
6	5	ralrimivw 2581	. . . 4 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → ∀𝑦 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)
7		dmopab3 4896	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵 ↔ dom {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} = 𝐴)
8	6, 7	sylib 122	. . 3 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → dom {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} = 𝐴)
9	4, 8	eqtrid 2251	. 2 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → dom (𝐴 × 𝐵) = 𝐴)
10	2, 9	sylbi 121	1 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → dom (𝐴 × 𝐵) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ∀wral 2485  {copab 4108   × cxp 4677  dom cdm 4679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-v 2775  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-br 4048  df-opab 4110  df-xp 4685  df-dm 4689

This theorem is referenced by:  rnxpm  5117  ssxpbm  5123  ssxp1  5124  xpexr2m  5129  relrelss  5214  unixpm  5223  exmidfodomrlemim  7316  pwsbas  13168  imasaddfnlemg  13190