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Theorem 1st2nd2 5853
Description: Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.)
Assertion
Ref Expression
1st2nd2 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)

Proof of Theorem 1st2nd2
StepHypRef Expression
1 elxp6 5848 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
21simplbi 268 1 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  cop 3420   × cxp 4390  cfv 4953  1st c1st 5817  2nd c2nd 5818
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2826  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-mpt 3862  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-iota 4918  df-fun 4955  df-fv 4961  df-1st 5819  df-2nd 5820
This theorem is referenced by:  xpopth  5854  eqop  5855  2nd1st  5858  1st2nd  5859  dfplpq2  6642  dfmpq2  6643  enqbreq2  6645  enqdc1  6650  preqlu  6760  prop  6763  elnp1st2nd  6764  cauappcvgprlemladd  6946  elreal2  7097  cnref1o  8850  frecuzrdgrrn  9526  frec2uzrdg  9527  frecuzrdgrcl  9528  frecuzrdgsuc  9532  frecuzrdgrclt  9533  frecuzrdgg  9534  frecuzrdgdomlem  9535  frecuzrdgfunlem  9537  frecuzrdgsuctlem  9541  iseqvalt  9568  eucalgval  10627  eucalginv  10629  eucalglt  10630  eucialg  10632  sqpweven  10744  2sqpwodd  10745  qnumdenbi  10761
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