Theorem 1st2nd2 5743

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 5738	. 2 ⊢ (A ∈ (B × 𝐶) ↔ (A = ⟨(1st ‘A), (2nd ‘A)⟩ ∧ ((1st ‘A) ∈ B ∧ (2nd ‘A) ∈ 𝐶)))
2	1	simplbi 259	1 ⊢ (A ∈ (B × 𝐶) → A = ⟨(1st ‘A), (2nd ‘A)⟩)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  ⟨cop 3370   × cxp 4286  ‘cfv 4845  1^st c1st 5707  2^nd c2nd 5708

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fv 4853  df-1st 5709  df-2nd 5710

This theorem is referenced by:  xpopth  5744  eqop  5745  2nd1st  5748  1st2nd  5749  dfplpq2  6338  dfmpq2  6339  enqbreq2  6341  enqdc1  6346  preqlu  6455  prop  6458  elnp1st2nd  6459  cauappcvgprlemladd  6630  elreal2  6728  cnref1o  8357  frecuzrdgrrn  8875