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Theorem elxp6 5796
 Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4808. (Contributed by NM, 9-Oct-2004.)
Assertion
Ref Expression
elxp6 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))

Proof of Theorem elxp6
StepHypRef Expression
1 elex 2566 . 2 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V)
2 opexg 3964 . . . 4 (((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ V)
32adantl 262 . . 3 ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ V)
4 eleq1 2100 . . . 4 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → (𝐴 ∈ V ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ V))
54adantr 261 . . 3 ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → (𝐴 ∈ V ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ V))
63, 5mpbird 156 . 2 ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → 𝐴 ∈ V)
7 1stvalg 5769 . . . . . 6 (𝐴 ∈ V → (1st𝐴) = dom {𝐴})
8 2ndvalg 5770 . . . . . 6 (𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
97, 8opeq12d 3557 . . . . 5 (𝐴 ∈ V → ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨ dom {𝐴}, ran {𝐴}⟩)
109eqeq2d 2051 . . . 4 (𝐴 ∈ V → (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ↔ 𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩))
117eleq1d 2106 . . . . 5 (𝐴 ∈ V → ((1st𝐴) ∈ 𝐵 dom {𝐴} ∈ 𝐵))
128eleq1d 2106 . . . . 5 (𝐴 ∈ V → ((2nd𝐴) ∈ 𝐶 ran {𝐴} ∈ 𝐶))
1311, 12anbi12d 442 . . . 4 (𝐴 ∈ V → (((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶) ↔ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
1410, 13anbi12d 442 . . 3 (𝐴 ∈ V → ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶))))
15 elxp4 4808 . . 3 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
1614, 15syl6rbbr 188 . 2 (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))))
171, 6, 16pm5.21nii 620 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  Vcvv 2557  {csn 3375  ⟨cop 3378  ∪ cuni 3580   × cxp 4343  dom cdm 4345  ran crn 4346  ‘cfv 4902  1st c1st 5765  2nd c2nd 5766 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-sbc 2765  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-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-1st 5767  df-2nd 5768 This theorem is referenced by:  elxp7  5797  eqopi  5798  1st2nd2  5801
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