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Theorem elxp6 5897
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4884. (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 2624 . 2 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V)
2 opexg 4029 . . . 4 (((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ V)
32adantl 271 . . 3 ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ V)
4 eleq1 2147 . . . 4 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → (𝐴 ∈ V ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ V))
54adantr 270 . . 3 ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → (𝐴 ∈ V ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ V))
63, 5mpbird 165 . 2 ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → 𝐴 ∈ V)
7 1stvalg 5870 . . . . . 6 (𝐴 ∈ V → (1st𝐴) = dom {𝐴})
8 2ndvalg 5871 . . . . . 6 (𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
97, 8opeq12d 3613 . . . . 5 (𝐴 ∈ V → ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨ dom {𝐴}, ran {𝐴}⟩)
109eqeq2d 2096 . . . 4 (𝐴 ∈ V → (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ↔ 𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩))
117eleq1d 2153 . . . . 5 (𝐴 ∈ V → ((1st𝐴) ∈ 𝐵 dom {𝐴} ∈ 𝐵))
128eleq1d 2153 . . . . 5 (𝐴 ∈ V → ((2nd𝐴) ∈ 𝐶 ran {𝐴} ∈ 𝐶))
1311, 12anbi12d 457 . . . 4 (𝐴 ∈ V → (((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶) ↔ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
1410, 13anbi12d 457 . . 3 (𝐴 ∈ V → ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶))))
15 elxp4 4884 . . 3 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
1614, 15syl6rbbr 197 . 2 (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))))
171, 6, 16pm5.21nii 653 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1287  wcel 1436  Vcvv 2615  {csn 3431  cop 3434   cuni 3636   × cxp 4409  dom cdm 4411  ran crn 4412  cfv 4981  1st c1st 5866  2nd c2nd 5867
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-mpt 3876  df-id 4094  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-iota 4946  df-fun 4983  df-fv 4989  df-1st 5868  df-2nd 5869
This theorem is referenced by:  elxp7  5898  eqopi  5899  1st2nd2  5902  eldju2ndl  6707  eldju2ndr  6708  qredeu  10954  qnumdencl  11040
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