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Theorem xpopth 7192
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 7190 . . 3 (𝐴 ∈ (𝐶 × 𝐷) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 1st2nd2 7190 . . 3 (𝐵 ∈ (𝑅 × 𝑆) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
31, 2eqeqan12d 2636 . 2 ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (𝐴 = 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩))
4 fvex 6188 . . 3 (1st𝐴) ∈ V
5 fvex 6188 . . 3 (2nd𝐴) ∈ V
64, 5opth 4935 . 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 1481  wcel 1988  cop 4174   × cxp 5102  cfv 5876  1st c1st 7151  2nd c2nd 7152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fv 5884  df-1st 7153  df-2nd 7154
This theorem is referenced by:  fseqdom  8834  iundom2g  9347  mdetunilem9  20407  txhaus  21431  fsumvma  24919  wlkeq  26510  disjxpin  29373  poimirlem4  33384  poimirlem13  33393  poimirlem14  33394  poimirlem22  33402  poimirlem26  33406  poimirlem27  33407  rmxypairf1o  37295
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