Theorem 1st2nd2 6338

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 6332	. 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 1397   ∈ wcel 2202  ⟨cop 3672   × cxp 4723  ‘cfv 5326  1^st c1st 6301  2^nd c2nd 6302

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fv 5334  df-1st 6303  df-2nd 6304

This theorem is referenced by:  xpopth  6339  eqop  6340  2nd1st  6343  1st2nd  6344  xpmapenlem  7035  opabfi  7132  djuf1olem  7252  exmidapne  7479  dfplpq2  7574  dfmpq2  7575  enqbreq2  7577  enqdc1  7582  preqlu  7692  prop  7695  elnp1st2nd  7696  cauappcvgprlemladd  7878  elreal2  8050  cnref1o  9885  frecuzrdgrrn  10671  frec2uzrdg  10672  frecuzrdgrcl  10673  frecuzrdgsuc  10677  frecuzrdgrclt  10678  frecuzrdgg  10679  frecuzrdgdomlem  10680  frecuzrdgfunlem  10682  frecuzrdgsuctlem  10686  seq3val  10723  seqvalcd  10724  eucalgval  12644  eucalginv  12646  eucalglt  12647  eucalg  12649  sqpweven  12765  2sqpwodd  12766  qnumdenbi  12782  xpsff1o  13450  tx1cn  15012  tx2cn  15013  txdis  15020  psmetxrge0  15075  xmetxpbl  15251