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

Theorem opreuopreu 8058
Description: There is a unique ordered pair fulfilling a wff iff its components fulfil a corresponding wff. (Contributed by AV, 2-Jul-2023.)
Hypothesis
Ref Expression
opreuopreu.a ((𝑎 = (1st𝑝) ∧ 𝑏 = (2nd𝑝)) → (𝜓𝜑))
Assertion
Ref Expression
opreuopreu (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑝   𝐵,𝑎,𝑏,𝑝   𝜑,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑝)   𝜓(𝑝,𝑎,𝑏)

Proof of Theorem opreuopreu
StepHypRef Expression
1 elxpi 5711 . . . 4 (𝑝 ∈ (𝐴 × 𝐵) → ∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)))
2 simprl 771 . . . . . . 7 ((𝜑 ∧ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵))) → 𝑝 = ⟨𝑎, 𝑏⟩)
3 vex 3482 . . . . . . . . . . . . . . 15 𝑎 ∈ V
4 vex 3482 . . . . . . . . . . . . . . 15 𝑏 ∈ V
53, 4op1st 8021 . . . . . . . . . . . . . 14 (1st ‘⟨𝑎, 𝑏⟩) = 𝑎
65eqcomi 2744 . . . . . . . . . . . . 13 𝑎 = (1st ‘⟨𝑎, 𝑏⟩)
73, 4op2nd 8022 . . . . . . . . . . . . . 14 (2nd ‘⟨𝑎, 𝑏⟩) = 𝑏
87eqcomi 2744 . . . . . . . . . . . . 13 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩)
96, 8pm3.2i 470 . . . . . . . . . . . 12 (𝑎 = (1st ‘⟨𝑎, 𝑏⟩) ∧ 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩))
10 fveq2 6907 . . . . . . . . . . . . . 14 (𝑝 = ⟨𝑎, 𝑏⟩ → (1st𝑝) = (1st ‘⟨𝑎, 𝑏⟩))
1110eqeq2d 2746 . . . . . . . . . . . . 13 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑎 = (1st𝑝) ↔ 𝑎 = (1st ‘⟨𝑎, 𝑏⟩)))
12 fveq2 6907 . . . . . . . . . . . . . 14 (𝑝 = ⟨𝑎, 𝑏⟩ → (2nd𝑝) = (2nd ‘⟨𝑎, 𝑏⟩))
1312eqeq2d 2746 . . . . . . . . . . . . 13 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑏 = (2nd𝑝) ↔ 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩)))
1411, 13anbi12d 632 . . . . . . . . . . . 12 (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝑎 = (1st𝑝) ∧ 𝑏 = (2nd𝑝)) ↔ (𝑎 = (1st ‘⟨𝑎, 𝑏⟩) ∧ 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩))))
159, 14mpbiri 258 . . . . . . . . . . 11 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑎 = (1st𝑝) ∧ 𝑏 = (2nd𝑝)))
16 opreuopreu.a . . . . . . . . . . 11 ((𝑎 = (1st𝑝) ∧ 𝑏 = (2nd𝑝)) → (𝜓𝜑))
1715, 16syl 17 . . . . . . . . . 10 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜓𝜑))
1817biimprd 248 . . . . . . . . 9 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑𝜓))
1918adantr 480 . . . . . . . 8 ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) → (𝜑𝜓))
2019impcom 407 . . . . . . 7 ((𝜑 ∧ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵))) → 𝜓)
212, 20jca 511 . . . . . 6 ((𝜑 ∧ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵))) → (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓))
2221ex 412 . . . . 5 (𝜑 → ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) → (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓)))
23222eximdv 1917 . . . 4 (𝜑 → (∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) → ∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓)))
241, 23syl5com 31 . . 3 (𝑝 ∈ (𝐴 × 𝐵) → (𝜑 → ∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓)))
2517biimpa 476 . . . . 5 ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓) → 𝜑)
2625a1i 11 . . . 4 (𝑝 ∈ (𝐴 × 𝐵) → ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓) → 𝜑))
2726exlimdvv 1932 . . 3 (𝑝 ∈ (𝐴 × 𝐵) → (∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓) → 𝜑))
2824, 27impbid 212 . 2 (𝑝 ∈ (𝐴 × 𝐵) → (𝜑 ↔ ∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓)))
2928reubiia 3385 1 (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  ∃!wreu 3376  cop 4637   × cxp 5687  cfv 6563  1st c1st 8011  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-2nd 8014
This theorem is referenced by:  2sqreuopb  27527
  Copyright terms: Public domain W3C validator