Theorem opreuopreu 8017

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 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)) → (𝜓 ↔ 𝜑))

Assertion

Ref	Expression
opreuopreu	⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)∃𝑎∃𝑏(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

Step	Hyp	Ref	Expression
1		elxpi 5698	. . . 4 ⊢ (𝑝 ∈ (𝐴 × 𝐵) → ∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)))
2		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵))) → 𝑝 = ⟨𝑎, 𝑏⟩)
3		vex 3479	. . . . . . . . . . . . . . 15 ⊢ 𝑎 ∈ V
4		vex 3479	. . . . . . . . . . . . . . 15 ⊢ 𝑏 ∈ V
5	3, 4	op1st 7980	. . . . . . . . . . . . . 14 ⊢ (1st ‘⟨𝑎, 𝑏⟩) = 𝑎
6	5	eqcomi 2742	. . . . . . . . . . . . 13 ⊢ 𝑎 = (1st ‘⟨𝑎, 𝑏⟩)
7	3, 4	op2nd 7981	. . . . . . . . . . . . . 14 ⊢ (2nd ‘⟨𝑎, 𝑏⟩) = 𝑏
8	7	eqcomi 2742	. . . . . . . . . . . . 13 ⊢ 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩)
9	6, 8	pm3.2i 472	. . . . . . . . . . . 12 ⊢ (𝑎 = (1st ‘⟨𝑎, 𝑏⟩) ∧ 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩))
10		fveq2 6889	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (1st ‘𝑝) = (1st ‘⟨𝑎, 𝑏⟩))
11	10	eqeq2d 2744	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑎 = (1st ‘𝑝) ↔ 𝑎 = (1st ‘⟨𝑎, 𝑏⟩)))
12		fveq2 6889	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (2nd ‘𝑝) = (2nd ‘⟨𝑎, 𝑏⟩))
13	12	eqeq2d 2744	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑏 = (2nd ‘𝑝) ↔ 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩)))
14	11, 13	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)) ↔ (𝑎 = (1st ‘⟨𝑎, 𝑏⟩) ∧ 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩))))
15	9, 14	mpbiri 258	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)))
16		opreuopreu.a	. . . . . . . . . . 11 ⊢ ((𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)) → (𝜓 ↔ 𝜑))
17	15, 16	syl 17	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜓 ↔ 𝜑))
18	17	biimprd 247	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 → 𝜓))
19	18	adantr 482	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝜑 → 𝜓))
20	19	impcom 409	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵))) → 𝜓)
21	2, 20	jca 513	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵))) → (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓))
22	21	ex 414	. . . . 5 ⊢ (𝜑 → ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓)))
23	22	2eximdv 1923	. . . 4 ⊢ (𝜑 → (∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → ∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓)))
24	1, 23	syl5com 31	. . 3 ⊢ (𝑝 ∈ (𝐴 × 𝐵) → (𝜑 → ∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓)))
25	17	biimpa 478	. . . . 5 ⊢ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓) → 𝜑)
26	25	a1i 11	. . . 4 ⊢ (𝑝 ∈ (𝐴 × 𝐵) → ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓) → 𝜑))
27	26	exlimdvv 1938	. . 3 ⊢ (𝑝 ∈ (𝐴 × 𝐵) → (∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓) → 𝜑))
28	24, 27	impbid 211	. 2 ⊢ (𝑝 ∈ (𝐴 × 𝐵) → (𝜑 ↔ ∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓)))
29	28	reubiia 3384	1 ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107  ∃!wreu 3375  ⟨cop 4634   × cxp 5674  ‘cfv 6541  1^st c1st 7970  2^nd c2nd 7971

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 5299  ax-nul 5306  ax-pr 5427  ax-un 7722

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-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6493  df-fun 6543  df-fv 6549  df-1st 7972  df-2nd 7973

This theorem is referenced by:  2sqreuopb  26961