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Theorem xpexr2m 4762
 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 4745 . 2 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
2 dmxpm 4555 . . . . . 6 (∃𝑏 𝑏𝐵 → dom (𝐴 × 𝐵) = 𝐴)
32adantl 262 . . . . 5 (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑏 𝑏𝐵) → dom (𝐴 × 𝐵) = 𝐴)
4 dmexg 4596 . . . . . 6 ((𝐴 × 𝐵) ∈ 𝐶 → dom (𝐴 × 𝐵) ∈ V)
54adantr 261 . . . . 5 (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑏 𝑏𝐵) → dom (𝐴 × 𝐵) ∈ V)
63, 5eqeltrrd 2115 . . . 4 (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑏 𝑏𝐵) → 𝐴 ∈ V)
7 rnxpm 4752 . . . . . 6 (∃𝑎 𝑎𝐴 → ran (𝐴 × 𝐵) = 𝐵)
87adantl 262 . . . . 5 (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑎 𝑎𝐴) → ran (𝐴 × 𝐵) = 𝐵)
9 rnexg 4597 . . . . . 6 ((𝐴 × 𝐵) ∈ 𝐶 → ran (𝐴 × 𝐵) ∈ V)
109adantr 261 . . . . 5 (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑎 𝑎𝐴) → ran (𝐴 × 𝐵) ∈ V)
118, 10eqeltrrd 2115 . . . 4 (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑎 𝑎𝐴) → 𝐵 ∈ V)
126, 11anim12dan 532 . . 3 (((𝐴 × 𝐵) ∈ 𝐶 ∧ (∃𝑏 𝑏𝐵 ∧ ∃𝑎 𝑎𝐴)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
1312ancom2s 500 . 2 (((𝐴 × 𝐵) ∈ 𝐶 ∧ (∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵)) → (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 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557   × cxp 4343  dom cdm 4345  ran crn 4346 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356 This theorem is referenced by: (None)
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