Theorem 1st2nd2 6327

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 6321	. 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 1395   ∈ wcel 2200  ⟨cop 3669   × cxp 4717  ‘cfv 5318  1^st c1st 6290  2^nd c2nd 6291

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fv 5326  df-1st 6292  df-2nd 6293

This theorem is referenced by:  xpopth  6328  eqop  6329  2nd1st  6332  1st2nd  6333  xpmapenlem  7018  opabfi  7108  djuf1olem  7228  exmidapne  7454  dfplpq2  7549  dfmpq2  7550  enqbreq2  7552  enqdc1  7557  preqlu  7667  prop  7670  elnp1st2nd  7671  cauappcvgprlemladd  7853  elreal2  8025  cnref1o  9854  frecuzrdgrrn  10638  frec2uzrdg  10639  frecuzrdgrcl  10640  frecuzrdgsuc  10644  frecuzrdgrclt  10645  frecuzrdgg  10646  frecuzrdgdomlem  10647  frecuzrdgfunlem  10649  frecuzrdgsuctlem  10653  seq3val  10690  seqvalcd  10691  eucalgval  12584  eucalginv  12586  eucalglt  12587  eucalg  12589  sqpweven  12705  2sqpwodd  12706  qnumdenbi  12722  xpsff1o  13390  tx1cn  14951  tx2cn  14952  txdis  14959  psmetxrge0  15014  xmetxpbl  15190