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Theorem elxp6 6074
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5033. (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 2700 . 2 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V)
2 opexg 4157 . . . 4 (((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ V)
32adantl 275 . . 3 ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ V)
4 eleq1 2203 . . . 4 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → (𝐴 ∈ V ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ V))
54adantr 274 . . 3 ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → (𝐴 ∈ V ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ V))
63, 5mpbird 166 . 2 ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → 𝐴 ∈ V)
7 1stvalg 6047 . . . . . 6 (𝐴 ∈ V → (1st𝐴) = dom {𝐴})
8 2ndvalg 6048 . . . . . 6 (𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
97, 8opeq12d 3720 . . . . 5 (𝐴 ∈ V → ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨ dom {𝐴}, ran {𝐴}⟩)
109eqeq2d 2152 . . . 4 (𝐴 ∈ V → (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ↔ 𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩))
117eleq1d 2209 . . . . 5 (𝐴 ∈ V → ((1st𝐴) ∈ 𝐵 dom {𝐴} ∈ 𝐵))
128eleq1d 2209 . . . . 5 (𝐴 ∈ V → ((2nd𝐴) ∈ 𝐶 ran {𝐴} ∈ 𝐶))
1311, 12anbi12d 465 . . . 4 (𝐴 ∈ V → (((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶) ↔ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
1410, 13anbi12d 465 . . 3 (𝐴 ∈ V → ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶))))
15 elxp4 5033 . . 3 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
1614, 15syl6rbbr 198 . 2 (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))))
171, 6, 16pm5.21nii 694 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1332  wcel 1481  Vcvv 2689  {csn 3531  cop 3534   cuni 3743   × cxp 4544  dom cdm 4546  ran crn 4547  cfv 5130  1st c1st 6043  2nd c2nd 6044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-iota 5095  df-fun 5132  df-fv 5138  df-1st 6045  df-2nd 6046
This theorem is referenced by:  elxp7  6075  oprssdmm  6076  eqopi  6077  1st2nd2  6080  eldju2ndl  6964  eldju2ndr  6965  qredeu  11812  qnumdencl  11899  tx1cn  12475  tx2cn  12476  psmetxrge0  12538  xmetxpbl  12714
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