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Theorem elxp6 6148
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5098. (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 2741 . 2 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V)
2 opexg 4213 . . . 4 (((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ V)
32adantl 275 . . 3 ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ V)
4 eleq1 2233 . . . 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 elxp4 5098 . . 3 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
8 1stvalg 6121 . . . . . 6 (𝐴 ∈ V → (1st𝐴) = dom {𝐴})
9 2ndvalg 6122 . . . . . 6 (𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
108, 9opeq12d 3773 . . . . 5 (𝐴 ∈ V → ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨ dom {𝐴}, ran {𝐴}⟩)
1110eqeq2d 2182 . . . 4 (𝐴 ∈ V → (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ↔ 𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩))
128eleq1d 2239 . . . . 5 (𝐴 ∈ V → ((1st𝐴) ∈ 𝐵 dom {𝐴} ∈ 𝐵))
139eleq1d 2239 . . . . 5 (𝐴 ∈ V → ((2nd𝐴) ∈ 𝐶 ran {𝐴} ∈ 𝐶))
1412, 13anbi12d 470 . . . 4 (𝐴 ∈ V → (((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶) ↔ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
1511, 14anbi12d 470 . . 3 (𝐴 ∈ V → ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶))))
167, 15bitr4id 198 . 2 (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))))
171, 6, 16pm5.21nii 699 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1348  wcel 2141  Vcvv 2730  {csn 3583  cop 3586   cuni 3796   × cxp 4609  dom cdm 4611  ran crn 4612  cfv 5198  1st c1st 6117  2nd c2nd 6118
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fv 5206  df-1st 6119  df-2nd 6120
This theorem is referenced by:  elxp7  6149  oprssdmm  6150  eqopi  6151  1st2nd2  6154  eldju2ndl  7049  eldju2ndr  7050  qredeu  12051  qnumdencl  12141  tx1cn  13063  tx2cn  13064  psmetxrge0  13126  xmetxpbl  13302
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