Theorem dmxpm 4498

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 2097	. . 3 ⊢ (x = z → (x ∈ B ↔ z ∈ B))
2	1	cbvexv 1792	. 2 ⊢ (∃x x ∈ B ↔ ∃z z ∈ B)
3		df-xp 4294	. . . 4 ⊢ (A × B) = {⟨y, z⟩ ∣ (y ∈ A ∧ z ∈ B)}
4	3	dmeqi 4479	. . 3 ⊢ dom (A × B) = dom {⟨y, z⟩ ∣ (y ∈ A ∧ z ∈ B)}
5		id 19	. . . . 5 ⊢ (∃z z ∈ B → ∃z z ∈ B)
6	5	ralrimivw 2387	. . . 4 ⊢ (∃z z ∈ B → ∀y ∈ A ∃z z ∈ B)
7		dmopab3 4491	. . . 4 ⊢ (∀y ∈ A ∃z z ∈ B ↔ dom {⟨y, z⟩ ∣ (y ∈ A ∧ z ∈ B)} = A)
8	6, 7	sylib 127	. . 3 ⊢ (∃z z ∈ B → dom {⟨y, z⟩ ∣ (y ∈ A ∧ z ∈ B)} = A)
9	4, 8	syl5eq 2081	. 2 ⊢ (∃z z ∈ B → dom (A × B) = A)
10	2, 9	sylbi 114	1 ⊢ (∃x x ∈ B → dom (A × B) = A)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∀wral 2300  {copab 3808   × cxp 4286  dom cdm 4288

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-dm 4298

This theorem is referenced by:  dmxpinm  4499  xpid11m  4500  rnxpm  4695  ssxpbm  4699  ssxp1  4700  xpexr2m  4705  relrelss  4787  unixpm  4796  xpiderm  6113