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Theorem dmxpm 4497
 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.
StepHypRef Expression
1 eleq1 2097 . . 3 (x = z → (x Bz B))
21cbvexv 1792 . 2 (x x Bz z B)
3 df-xp 4293 . . . 4 (A × B) = {⟨y, z⟩ ∣ (y A z B)}
43dmeqi 4478 . . 3 dom (A × B) = dom {⟨y, z⟩ ∣ (y A z B)}
5 id 19 . . . . 5 (z z Bz z B)
65ralrimivw 2387 . . . 4 (z z By A z z B)
7 dmopab3 4490 . . . 4 (y A z z B ↔ dom {⟨y, z⟩ ∣ (y A z B)} = A)
86, 7sylib 127 . . 3 (z z B → dom {⟨y, z⟩ ∣ (y A z B)} = A)
94, 8syl5eq 2081 . 2 (z z B → dom (A × B) = A)
102, 9sylbi 114 1 (x x B → dom (A × B) = A)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∀wral 2300  {copab 3807   × cxp 4285  dom cdm 4287 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-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934 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 3352  df-sn 3372  df-pr 3373  df-op 3375  df-br 3755  df-opab 3809  df-xp 4293  df-dm 4297 This theorem is referenced by:  dmxpinm  4498  xpid11m  4499  rnxpm  4694  ssxpbm  4698  ssxp1  4699  xpexr2m  4704  relrelss  4786  unixpm  4795  xpiderm  6106
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