Theorem 1st2nd2 6175

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 6169	. 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 1353   ∈ wcel 2148  ⟨cop 3595   × cxp 4624  ‘cfv 5216  1^st c1st 6138  2^nd c2nd 6139

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fv 5224  df-1st 6140  df-2nd 6141

This theorem is referenced by:  xpopth  6176  eqop  6177  2nd1st  6180  1st2nd  6181  xpmapenlem  6848  djuf1olem  7051  exmidapne  7258  dfplpq2  7352  dfmpq2  7353  enqbreq2  7355  enqdc1  7360  preqlu  7470  prop  7473  elnp1st2nd  7474  cauappcvgprlemladd  7656  elreal2  7828  cnref1o  9649  frecuzrdgrrn  10407  frec2uzrdg  10408  frecuzrdgrcl  10409  frecuzrdgsuc  10413  frecuzrdgrclt  10414  frecuzrdgg  10415  frecuzrdgdomlem  10416  frecuzrdgfunlem  10418  frecuzrdgsuctlem  10422  seq3val  10457  seqvalcd  10458  eucalgval  12053  eucalginv  12055  eucalglt  12056  eucalg  12058  sqpweven  12174  2sqpwodd  12175  qnumdenbi  12191  xpsff1o  12767  tx1cn  13739  tx2cn  13740  txdis  13747  psmetxrge0  13802  xmetxpbl  13978