Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpopth Structured version   Visualization version   GIF version

Theorem xpopth 7167
 Description: An ordered pair theorem for members of Cartesian products. (Contributed by NM, 20-Jun-2007.)
Assertion
Ref Expression
xpopth ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ 𝐴 = 𝐵))

Proof of Theorem xpopth
StepHypRef Expression
1 1st2nd2 7165 . . 3 (𝐴 ∈ (𝐶 × 𝐷) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 1st2nd2 7165 . . 3 (𝐵 ∈ (𝑅 × 𝑆) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
31, 2eqeqan12d 2637 . 2 ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (𝐴 = 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩))
4 fvex 6168 . . 3 (1st𝐴) ∈ V
5 fvex 6168 . . 3 (2nd𝐴) ∈ V
64, 5opth 4915 . 2 (⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)))
73, 6syl6rbb 277 1 ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ 𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ⟨cop 4161   × cxp 5082  ‘cfv 5857  1st c1st 7126  2nd c2nd 7127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fv 5865  df-1st 7128  df-2nd 7129 This theorem is referenced by:  fseqdom  8809  iundom2g  9322  mdetunilem9  20366  txhaus  21390  fsumvma  24872  wlkeq  26433  disjxpin  29287  poimirlem4  33084  poimirlem13  33093  poimirlem14  33094  poimirlem22  33102  poimirlem26  33106  poimirlem27  33107  rmxypairf1o  36995
 Copyright terms: Public domain W3C validator