Theorem 1st2nd2 6123

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 6117	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))
2	1	simplbi 272	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1335   ∈ wcel 2128  ⟨cop 3563   × cxp 4584  ‘cfv 5170  1^st c1st 6086  2^nd c2nd 6087

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-iota 5135  df-fun 5172  df-fv 5178  df-1st 6088  df-2nd 6089

This theorem is referenced by:  xpopth  6124  eqop  6125  2nd1st  6128  1st2nd  6129  xpmapenlem  6794  djuf1olem  6997  dfplpq2  7274  dfmpq2  7275  enqbreq2  7277  enqdc1  7282  preqlu  7392  prop  7395  elnp1st2nd  7396  cauappcvgprlemladd  7578  elreal2  7750  cnref1o  9559  frecuzrdgrrn  10307  frec2uzrdg  10308  frecuzrdgrcl  10309  frecuzrdgsuc  10313  frecuzrdgrclt  10314  frecuzrdgg  10315  frecuzrdgdomlem  10316  frecuzrdgfunlem  10318  frecuzrdgsuctlem  10322  seq3val  10357  seqvalcd  10358  eucalgval  11930  eucalginv  11932  eucalglt  11933  eucalg  11935  sqpweven  12049  2sqpwodd  12050  qnumdenbi  12066  tx1cn  12669  tx2cn  12670  txdis  12677  psmetxrge0  12732  xmetxpbl  12908