Theorem dmxpm 4831

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 2233	. . 3 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
2	1	cbvexv 1911	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵)
3		df-xp 4617	. . . 4 ⊢ (𝐴 × 𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}
4	3	dmeqi 4812	. . 3 ⊢ dom (𝐴 × 𝐵) = dom {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}
5		id 19	. . . . 5 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → ∃𝑧 𝑧 ∈ 𝐵)
6	5	ralrimivw 2544	. . . 4 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → ∀𝑦 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)
7		dmopab3 4824	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵 ↔ dom {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} = 𝐴)
8	6, 7	sylib 121	. . 3 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → dom {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} = 𝐴)
9	4, 8	eqtrid 2215	. 2 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → dom (𝐴 × 𝐵) = 𝐴)
10	2, 9	sylbi 120	1 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → dom (𝐴 × 𝐵) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348  ∃wex 1485   ∈ wcel 2141  ∀wral 2448  {copab 4049   × cxp 4609  dom cdm 4611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-dm 4621

This theorem is referenced by:  rnxpm  5040  ssxpbm  5046  ssxp1  5047  xpexr2m  5052  relrelss  5137  unixpm  5146  exmidfodomrlemim  7178