Theorem 1st2nd2 6176

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 6170	. 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 3596   × cxp 4625  ‘cfv 5217  1^st c1st 6139  2^nd c2nd 6140

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 4122  ax-pow 4175  ax-pr 4210  ax-un 4434

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 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fv 5225  df-1st 6141  df-2nd 6142

This theorem is referenced by:  xpopth  6177  eqop  6178  2nd1st  6181  1st2nd  6182  xpmapenlem  6849  djuf1olem  7052  exmidapne  7259  dfplpq2  7353  dfmpq2  7354  enqbreq2  7356  enqdc1  7361  preqlu  7471  prop  7474  elnp1st2nd  7475  cauappcvgprlemladd  7657  elreal2  7829  cnref1o  9650  frecuzrdgrrn  10408  frec2uzrdg  10409  frecuzrdgrcl  10410  frecuzrdgsuc  10414  frecuzrdgrclt  10415  frecuzrdgg  10416  frecuzrdgdomlem  10417  frecuzrdgfunlem  10419  frecuzrdgsuctlem  10423  seq3val  10458  seqvalcd  10459  eucalgval  12054  eucalginv  12056  eucalglt  12057  eucalg  12059  sqpweven  12175  2sqpwodd  12176  qnumdenbi  12192  xpsff1o  12768  tx1cn  13772  tx2cn  13773  txdis  13780  psmetxrge0  13835  xmetxpbl  14011