Theorem 1st2nd2 6228

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 6222	. 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 1364   ∈ wcel 2164  ⟨cop 3621   × cxp 4657  ‘cfv 5254  1^st c1st 6191  2^nd c2nd 6192

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fv 5262  df-1st 6193  df-2nd 6194

This theorem is referenced by:  xpopth  6229  eqop  6230  2nd1st  6233  1st2nd  6234  xpmapenlem  6905  opabfi  6992  djuf1olem  7112  exmidapne  7320  dfplpq2  7414  dfmpq2  7415  enqbreq2  7417  enqdc1  7422  preqlu  7532  prop  7535  elnp1st2nd  7536  cauappcvgprlemladd  7718  elreal2  7890  cnref1o  9716  frecuzrdgrrn  10479  frec2uzrdg  10480  frecuzrdgrcl  10481  frecuzrdgsuc  10485  frecuzrdgrclt  10486  frecuzrdgg  10487  frecuzrdgdomlem  10488  frecuzrdgfunlem  10490  frecuzrdgsuctlem  10494  seq3val  10531  seqvalcd  10532  eucalgval  12192  eucalginv  12194  eucalglt  12195  eucalg  12197  sqpweven  12313  2sqpwodd  12314  qnumdenbi  12330  xpsff1o  12932  tx1cn  14437  tx2cn  14438  txdis  14445  psmetxrge0  14500  xmetxpbl  14676