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Theorem opreuopreu 8019
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 5691 . . . 4 (𝑝 ∈ (𝐴 × 𝐵) → ∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵)))
2 simprl 768 . . . . . . 7 ((𝜑 ∧ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐵))) → 𝑝 = ⟨𝑎, 𝑏⟩)
3 vex 3472 . . . . . . . . . . . . . . 15 𝑎 ∈ V
4 vex 3472 . . . . . . . . . . . . . . 15 𝑏 ∈ V
53, 4op1st 7982 . . . . . . . . . . . . . 14 (1st ‘⟨𝑎, 𝑏⟩) = 𝑎
65eqcomi 2735 . . . . . . . . . . . . 13 𝑎 = (1st ‘⟨𝑎, 𝑏⟩)
73, 4op2nd 7983 . . . . . . . . . . . . . 14 (2nd ‘⟨𝑎, 𝑏⟩) = 𝑏
87eqcomi 2735 . . . . . . . . . . . . 13 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩)
96, 8pm3.2i 470 . . . . . . . . . . . 12 (𝑎 = (1st ‘⟨𝑎, 𝑏⟩) ∧ 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩))
10 fveq2 6885 . . . . . . . . . . . . . 14 (𝑝 = ⟨𝑎, 𝑏⟩ → (1st𝑝) = (1st ‘⟨𝑎, 𝑏⟩))
1110eqeq2d 2737 . . . . . . . . . . . . 13 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑎 = (1st𝑝) ↔ 𝑎 = (1st ‘⟨𝑎, 𝑏⟩)))
12 fveq2 6885 . . . . . . . . . . . . . 14 (𝑝 = ⟨𝑎, 𝑏⟩ → (2nd𝑝) = (2nd ‘⟨𝑎, 𝑏⟩))
1312eqeq2d 2737 . . . . . . . . . . . . 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 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 247 . . . . . . . . 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 1914 . . . 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 1929 . . 3 (𝑝 ∈ (𝐴 × 𝐵) → (∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓) → 𝜑))
2824, 27impbid 211 . 2 (𝑝 ∈ (𝐴 × 𝐵) → (𝜑 ↔ ∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓)))
2928reubiia 3377 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 395   = wceq 1533  wex 1773  wcel 2098  ∃!wreu 3368  cop 4629   × cxp 5667  cfv 6537  1st c1st 7972  2nd c2nd 7973
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fv 6545  df-1st 7974  df-2nd 7975
This theorem is referenced by:  2sqreuopb  27356
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