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Theorem elxp6 5823
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4835. (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 2583 . 2 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V)
2 opexg 3991 . . . 4 (((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ V)
32adantl 266 . . 3 ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ V)
4 eleq1 2116 . . . 4 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → (𝐴 ∈ V ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ V))
54adantr 265 . . 3 ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → (𝐴 ∈ V ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ V))
63, 5mpbird 160 . 2 ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → 𝐴 ∈ V)
7 1stvalg 5796 . . . . . 6 (𝐴 ∈ V → (1st𝐴) = dom {𝐴})
8 2ndvalg 5797 . . . . . 6 (𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
97, 8opeq12d 3584 . . . . 5 (𝐴 ∈ V → ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨ dom {𝐴}, ran {𝐴}⟩)
109eqeq2d 2067 . . . 4 (𝐴 ∈ V → (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ↔ 𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩))
117eleq1d 2122 . . . . 5 (𝐴 ∈ V → ((1st𝐴) ∈ 𝐵 dom {𝐴} ∈ 𝐵))
128eleq1d 2122 . . . . 5 (𝐴 ∈ V → ((2nd𝐴) ∈ 𝐶 ran {𝐴} ∈ 𝐶))
1311, 12anbi12d 450 . . . 4 (𝐴 ∈ V → (((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶) ↔ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
1410, 13anbi12d 450 . . 3 (𝐴 ∈ V → ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶))))
15 elxp4 4835 . . 3 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
1614, 15syl6rbbr 192 . 2 (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))))
171, 6, 16pm5.21nii 630 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102   = wceq 1259  wcel 1409  Vcvv 2574  {csn 3402  cop 3405   cuni 3607   × cxp 4370  dom cdm 4372  ran crn 4373  cfv 4929  1st c1st 5792  2nd c2nd 5793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fv 4937  df-1st 5794  df-2nd 5795
This theorem is referenced by:  elxp7  5824  eqopi  5825  1st2nd2  5828
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