Theorem dmxpm 4920

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 2272	. . 3 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
2	1	cbvexv 1945	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵)
3		df-xp 4702	. . . 4 ⊢ (𝐴 × 𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}
4	3	dmeqi 4901	. . 3 ⊢ dom (𝐴 × 𝐵) = dom {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}
5		id 19	. . . . 5 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → ∃𝑧 𝑧 ∈ 𝐵)
6	5	ralrimivw 2584	. . . 4 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → ∀𝑦 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)
7		dmopab3 4913	. . . 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 2254	. 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 1375  ∃wex 1518   ∈ wcel 2180  ∀wral 2488  {copab 4123   × cxp 4694  dom cdm 4696

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-v 2781  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-br 4063  df-opab 4125  df-xp 4702  df-dm 4706

This theorem is referenced by:  rnxpm  5134  ssxpbm  5140  ssxp1  5141  xpexr2m  5146  relrelss  5231  unixpm  5240  exmidfodomrlemim  7347  pwsbas  13291  imasaddfnlemg  13313