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Theorem xpexr2m 4705
 Description: If a nonempty cross product is a set, so are both of its components. (Contributed by Jim Kingdon, 14-Dec-2018.)
Assertion
Ref Expression
xpexr2m (((A × B) 𝐶 x x (A × B)) → (A V B V))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   𝐶(x)

Proof of Theorem xpexr2m
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpm 4688 . 2 ((𝑎 𝑎 A 𝑏 𝑏 B) ↔ x x (A × B))
2 dmxpm 4498 . . . . . 6 (𝑏 𝑏 B → dom (A × B) = A)
32adantl 262 . . . . 5 (((A × B) 𝐶 𝑏 𝑏 B) → dom (A × B) = A)
4 dmexg 4539 . . . . . 6 ((A × B) 𝐶 → dom (A × B) V)
54adantr 261 . . . . 5 (((A × B) 𝐶 𝑏 𝑏 B) → dom (A × B) V)
63, 5eqeltrrd 2112 . . . 4 (((A × B) 𝐶 𝑏 𝑏 B) → A V)
7 rnxpm 4695 . . . . . 6 (𝑎 𝑎 A → ran (A × B) = B)
87adantl 262 . . . . 5 (((A × B) 𝐶 𝑎 𝑎 A) → ran (A × B) = B)
9 rnexg 4540 . . . . . 6 ((A × B) 𝐶 → ran (A × B) V)
109adantr 261 . . . . 5 (((A × B) 𝐶 𝑎 𝑎 A) → ran (A × B) V)
118, 10eqeltrrd 2112 . . . 4 (((A × B) 𝐶 𝑎 𝑎 A) → B V)
126, 11anim12dan 532 . . 3 (((A × B) 𝐶 (𝑏 𝑏 B 𝑎 𝑎 A)) → (A V B V))
1312ancom2s 500 . 2 (((A × B) 𝐶 (𝑎 𝑎 A 𝑏 𝑏 B)) → (A V B V))
141, 13sylan2br 272 1 (((A × B) 𝐶 x x (A × B)) → (A V B V))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551   × cxp 4286  dom cdm 4288  ran crn 4289 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-13 1401  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  ax-un 4136 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-rex 2306  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-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299 This theorem is referenced by: (None)
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