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Theorem opreuopreu 7728
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 5575 . . . 4 (𝑝 ∈ (𝐴 × 𝐵) → ∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)))
2 simprl 767 . . . . . . 7 ((𝜑 ∧ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵))) → 𝑝 = ⟨𝑎, 𝑏⟩)
3 vex 3502 . . . . . . . . . . . . . . 15 𝑎 ∈ V
4 vex 3502 . . . . . . . . . . . . . . 15 𝑏 ∈ V
53, 4op1st 7691 . . . . . . . . . . . . . 14 (1st ‘⟨𝑎, 𝑏⟩) = 𝑎
65eqcomi 2833 . . . . . . . . . . . . 13 𝑎 = (1st ‘⟨𝑎, 𝑏⟩)
73, 4op2nd 7692 . . . . . . . . . . . . . 14 (2nd ‘⟨𝑎, 𝑏⟩) = 𝑏
87eqcomi 2833 . . . . . . . . . . . . 13 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩)
96, 8pm3.2i 471 . . . . . . . . . . . 12 (𝑎 = (1st ‘⟨𝑎, 𝑏⟩) ∧ 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩))
10 fveq2 6666 . . . . . . . . . . . . . 14 (𝑝 = ⟨𝑎, 𝑏⟩ → (1st𝑝) = (1st ‘⟨𝑎, 𝑏⟩))
1110eqeq2d 2835 . . . . . . . . . . . . 13 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑎 = (1st𝑝) ↔ 𝑎 = (1st ‘⟨𝑎, 𝑏⟩)))
12 fveq2 6666 . . . . . . . . . . . . . 14 (𝑝 = ⟨𝑎, 𝑏⟩ → (2nd𝑝) = (2nd ‘⟨𝑎, 𝑏⟩))
1312eqeq2d 2835 . . . . . . . . . . . . 13 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑏 = (2nd𝑝) ↔ 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩)))
1411, 13anbi12d 630 . . . . . . . . . . . 12 (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝑎 = (1st𝑝) ∧ 𝑏 = (2nd𝑝)) ↔ (𝑎 = (1st ‘⟨𝑎, 𝑏⟩) ∧ 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩))))
159, 14mpbiri 259 . . . . . . . . . . 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 249 . . . . . . . . 9 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑𝜓))
1918adantr 481 . . . . . . . 8 ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) → (𝜑𝜓))
2019impcom 408 . . . . . . 7 ((𝜑 ∧ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵))) → 𝜓)
212, 20jca 512 . . . . . 6 ((𝜑 ∧ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵))) → (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓))
2221ex 413 . . . . 5 (𝜑 → ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)) → (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓)))
23222eximdv 1913 . . . 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 477 . . . . 5 ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓) → 𝜑)
2625a1i 11 . . . 4 (𝑝 ∈ (𝐴 × 𝐵) → ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓) → 𝜑))
2726exlimdvv 1928 . . 3 (𝑝 ∈ (𝐴 × 𝐵) → (∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓) → 𝜑))
2824, 27impbid 213 . 2 (𝑝 ∈ (𝐴 × 𝐵) → (𝜑 ↔ ∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓)))
2928reubiia 3395 1 (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2106  ∃!wreu 3144  cop 4569   × cxp 5551  cfv 6351  1st c1st 7681  2nd c2nd 7682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-iota 6311  df-fun 6353  df-fv 6359  df-1st 7683  df-2nd 7684
This theorem is referenced by:  2sqreuopb  25958
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