Theorem 1st2nd2 6193

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 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)

Proof of Theorem 1st2nd2

Step	Hyp	Ref	Expression
1		elxp6 6187	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))
2	1	simplbi 274	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1363   ∈ wcel 2159  ⟨cop 3609   × cxp 4638  ‘cfv 5230  1^st c1st 6156  2^nd c2nd 6157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-v 2753  df-sbc 2977  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-opab 4079  df-mpt 4080  df-id 4307  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-iota 5192  df-fun 5232  df-fv 5238  df-1st 6158  df-2nd 6159

This theorem is referenced by:  xpopth  6194  eqop  6195  2nd1st  6198  1st2nd  6199  xpmapenlem  6866  djuf1olem  7069  exmidapne  7276  dfplpq2  7370  dfmpq2  7371  enqbreq2  7373  enqdc1  7378  preqlu  7488  prop  7491  elnp1st2nd  7492  cauappcvgprlemladd  7674  elreal2  7846  cnref1o  9667  frecuzrdgrrn  10425  frec2uzrdg  10426  frecuzrdgrcl  10427  frecuzrdgsuc  10431  frecuzrdgrclt  10432  frecuzrdgg  10433  frecuzrdgdomlem  10434  frecuzrdgfunlem  10436  frecuzrdgsuctlem  10440  seq3val  10475  seqvalcd  10476  eucalgval  12071  eucalginv  12073  eucalglt  12074  eucalg  12076  sqpweven  12192  2sqpwodd  12193  qnumdenbi  12209  xpsff1o  12790  tx1cn  14152  tx2cn  14153  txdis  14160  psmetxrge0  14215  xmetxpbl  14391