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Theorem xpexr2m 4826
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 4807 . 2 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
2 dmxpm 4614 . . . . . 6 (∃𝑏 𝑏𝐵 → dom (𝐴 × 𝐵) = 𝐴)
32adantl 271 . . . . 5 (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑏 𝑏𝐵) → dom (𝐴 × 𝐵) = 𝐴)
4 dmexg 4655 . . . . . 6 ((𝐴 × 𝐵) ∈ 𝐶 → dom (𝐴 × 𝐵) ∈ V)
54adantr 270 . . . . 5 (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑏 𝑏𝐵) → dom (𝐴 × 𝐵) ∈ V)
63, 5eqeltrrd 2160 . . . 4 (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑏 𝑏𝐵) → 𝐴 ∈ V)
7 rnxpm 4814 . . . . . 6 (∃𝑎 𝑎𝐴 → ran (𝐴 × 𝐵) = 𝐵)
87adantl 271 . . . . 5 (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑎 𝑎𝐴) → ran (𝐴 × 𝐵) = 𝐵)
9 rnexg 4656 . . . . . 6 ((𝐴 × 𝐵) ∈ 𝐶 → ran (𝐴 × 𝐵) ∈ V)
109adantr 270 . . . . 5 (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑎 𝑎𝐴) → ran (𝐴 × 𝐵) ∈ V)
118, 10eqeltrrd 2160 . . . 4 (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑎 𝑎𝐴) → 𝐵 ∈ V)
126, 11anim12dan 565 . . 3 (((𝐴 × 𝐵) ∈ 𝐶 ∧ (∃𝑏 𝑏𝐵 ∧ ∃𝑎 𝑎𝐴)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
1312ancom2s 531 . 2 (((𝐴 × 𝐵) ∈ 𝐶 ∧ (∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
141, 13sylan2br 282 1 (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wex 1422  wcel 1434  Vcvv 2612   × cxp 4399  dom cdm 4401  ran crn 4402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-xp 4407  df-rel 4408  df-cnv 4409  df-dm 4411  df-rn 4412
This theorem is referenced by: (None)
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