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Theorem xpopth 7446
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 7444 . . 3 (𝐴 ∈ (𝐶 × 𝐷) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 1st2nd2 7444 . . 3 (𝐵 ∈ (𝑅 × 𝑆) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
31, 2eqeqan12d 2819 . 2 ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (𝐴 = 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩))
4 fvex 6428 . . 3 (1st𝐴) ∈ V
5 fvex 6428 . . 3 (2nd𝐴) ∈ V
64, 5opth 5139 . 2 (⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)))
73, 6syl6rbb 280 1 ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  cop 4378   × cxp 5314  cfv 6105  1st c1st 7403  2nd c2nd 7404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2379  ax-ext 2781  ax-sep 4979  ax-nul 4987  ax-pow 5039  ax-pr 5101  ax-un 7187
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2593  df-eu 2611  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ral 3098  df-rex 3099  df-rab 3102  df-v 3391  df-sbc 3638  df-dif 3776  df-un 3778  df-in 3780  df-ss 3787  df-nul 4120  df-if 4282  df-sn 4373  df-pr 4375  df-op 4379  df-uni 4633  df-br 4848  df-opab 4910  df-mpt 4927  df-id 5224  df-xp 5322  df-rel 5323  df-cnv 5324  df-co 5325  df-dm 5326  df-rn 5327  df-iota 6068  df-fun 6107  df-fv 6113  df-1st 7405  df-2nd 7406
This theorem is referenced by:  fseqdom  9139  iundom2g  9654  mdetunilem9  20756  txhaus  21783  fsumvma  25294  wlkeq  26887  disjxpin  29922  poimirlem4  33906  poimirlem13  33915  poimirlem14  33916  poimirlem22  33924  poimirlem26  33928  poimirlem27  33929  rmxypairf1o  38265
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