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Theorem opreuopreu 7967
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 5656 . . . 4 (𝑝 ∈ (𝐴 × 𝐵) → ∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)))
2 simprl 770 . . . . . . 7 ((𝜑 ∧ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵))) → 𝑝 = ⟨𝑎, 𝑏⟩)
3 vex 3450 . . . . . . . . . . . . . . 15 𝑎 ∈ V
4 vex 3450 . . . . . . . . . . . . . . 15 𝑏 ∈ V
53, 4op1st 7930 . . . . . . . . . . . . . 14 (1st ‘⟨𝑎, 𝑏⟩) = 𝑎
65eqcomi 2746 . . . . . . . . . . . . 13 𝑎 = (1st ‘⟨𝑎, 𝑏⟩)
73, 4op2nd 7931 . . . . . . . . . . . . . 14 (2nd ‘⟨𝑎, 𝑏⟩) = 𝑏
87eqcomi 2746 . . . . . . . . . . . . 13 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩)
96, 8pm3.2i 472 . . . . . . . . . . . 12 (𝑎 = (1st ‘⟨𝑎, 𝑏⟩) ∧ 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩))
10 fveq2 6843 . . . . . . . . . . . . . 14 (𝑝 = ⟨𝑎, 𝑏⟩ → (1st𝑝) = (1st ‘⟨𝑎, 𝑏⟩))
1110eqeq2d 2748 . . . . . . . . . . . . 13 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑎 = (1st𝑝) ↔ 𝑎 = (1st ‘⟨𝑎, 𝑏⟩)))
12 fveq2 6843 . . . . . . . . . . . . . 14 (𝑝 = ⟨𝑎, 𝑏⟩ → (2nd𝑝) = (2nd ‘⟨𝑎, 𝑏⟩))
1312eqeq2d 2748 . . . . . . . . . . . . 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 482 . . . . . . . 8 ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) → (𝜑𝜓))
2019impcom 409 . . . . . . 7 ((𝜑 ∧ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵))) → 𝜓)
212, 20jca 513 . . . . . 6 ((𝜑 ∧ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵))) → (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓))
2221ex 414 . . . . 5 (𝜑 → ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) → (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓)))
23222eximdv 1923 . . . 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 478 . . . . 5 ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓) → 𝜑)
2625a1i 11 . . . 4 (𝑝 ∈ (𝐴 × 𝐵) → ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓) → 𝜑))
2726exlimdvv 1938 . . 3 (𝑝 ∈ (𝐴 × 𝐵) → (∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓) → 𝜑))
2824, 27impbid 211 . 2 (𝑝 ∈ (𝐴 × 𝐵) → (𝜑 ↔ ∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓)))
2928reubiia 3361 1 (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  ∃!wreu 3352  cop 4593   × cxp 5632  cfv 6497  1st c1st 7920  2nd c2nd 7921
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 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-1st 7922  df-2nd 7923
This theorem is referenced by:  2sqreuopb  26819
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