MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opreu2reurex Structured version   Visualization version   GIF version

Theorem opreu2reurex 6294
Description: There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 24-Jun-2023.) (Revised by AV, 1-Jul-2023.)
Hypothesis
Ref Expression
opreu2reurex.a (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑𝜒))
Assertion
Ref Expression
opreu2reurex (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃!𝑎𝐴𝑏𝐵 𝜒 ∧ ∃!𝑏𝐵𝑎𝐴 𝜒))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑝   𝐵,𝑎,𝑏,𝑝   𝜑,𝑎,𝑏   𝜒,𝑝
Allowed substitution hints:   𝜑(𝑝)   𝜒(𝑎,𝑏)

Proof of Theorem opreu2reurex
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqcom 2740 . . . . . . . 8 (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩)
2 vex 3479 . . . . . . . . 9 𝑎 ∈ V
3 vex 3479 . . . . . . . . 9 𝑏 ∈ V
42, 3opth 5477 . . . . . . . 8 (⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑎 = 𝑥𝑏 = 𝑦))
51, 4bitri 275 . . . . . . 7 (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ↔ (𝑎 = 𝑥𝑏 = 𝑦))
65imbi2i 336 . . . . . 6 ((𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩) ↔ (𝜒 → (𝑎 = 𝑥𝑏 = 𝑦)))
76a1i 11 . . . . 5 (((𝑥𝐴𝑦𝐵) ∧ (𝑎𝐴𝑏𝐵)) → ((𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩) ↔ (𝜒 → (𝑎 = 𝑥𝑏 = 𝑦))))
872ralbidva 3217 . . . 4 ((𝑥𝐴𝑦𝐵) → (∀𝑎𝐴𝑏𝐵 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩) ↔ ∀𝑎𝐴𝑏𝐵 (𝜒 → (𝑎 = 𝑥𝑏 = 𝑦))))
982rexbiia 3216 . . 3 (∃𝑥𝐴𝑦𝐵𝑎𝐴𝑏𝐵 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩) ↔ ∃𝑥𝐴𝑦𝐵𝑎𝐴𝑏𝐵 (𝜒 → (𝑎 = 𝑥𝑏 = 𝑦)))
109anbi2i 624 . 2 ((∃𝑎𝐴𝑏𝐵 𝜒 ∧ ∃𝑥𝐴𝑦𝐵𝑎𝐴𝑏𝐵 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)) ↔ (∃𝑎𝐴𝑏𝐵 𝜒 ∧ ∃𝑥𝐴𝑦𝐵𝑎𝐴𝑏𝐵 (𝜒 → (𝑎 = 𝑥𝑏 = 𝑦))))
11 opreu2reurex.a . . 3 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑𝜒))
1211reu3op 6292 . 2 (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃𝑎𝐴𝑏𝐵 𝜒 ∧ ∃𝑥𝐴𝑦𝐵𝑎𝐴𝑏𝐵 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
13 2reu4 4527 . 2 ((∃!𝑎𝐴𝑏𝐵 𝜒 ∧ ∃!𝑏𝐵𝑎𝐴 𝜒) ↔ (∃𝑎𝐴𝑏𝐵 𝜒 ∧ ∃𝑥𝐴𝑦𝐵𝑎𝐴𝑏𝐵 (𝜒 → (𝑎 = 𝑥𝑏 = 𝑦))))
1410, 12, 133bitr4i 303 1 (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃!𝑎𝐴𝑏𝐵 𝜒 ∧ ∃!𝑏𝐵𝑎𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  ∃!wreu 3375  cop 4635   × cxp 5675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-iun 5000  df-opab 5212  df-xp 5683  df-rel 5684
This theorem is referenced by:  opreu2reu  6295  2sqreuop  26965  2sqreuopnn  26966  2sqreuoplt  26967  2sqreuopltb  26968  2sqreuopnnlt  26969  2sqreuopnnltb  26970  opreu2reu1  31724
  Copyright terms: Public domain W3C validator