Theorem opreuopreu 7969

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 5641	. . . 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 3440	. . . . . . . . . . . . . . 15 ⊢ 𝑎 ∈ V
4		vex 3440	. . . . . . . . . . . . . . 15 ⊢ 𝑏 ∈ V
5	3, 4	op1st 7932	. . . . . . . . . . . . . 14 ⊢ (1st ‘⟨𝑎, 𝑏⟩) = 𝑎
6	5	eqcomi 2738	. . . . . . . . . . . . 13 ⊢ 𝑎 = (1st ‘⟨𝑎, 𝑏⟩)
7	3, 4	op2nd 7933	. . . . . . . . . . . . . 14 ⊢ (2nd ‘⟨𝑎, 𝑏⟩) = 𝑏
8	7	eqcomi 2738	. . . . . . . . . . . . 13 ⊢ 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩)
9	6, 8	pm3.2i 470	. . . . . . . . . . . 12 ⊢ (𝑎 = (1st ‘⟨𝑎, 𝑏⟩) ∧ 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩))
10		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (1st ‘𝑝) = (1st ‘⟨𝑎, 𝑏⟩))
11	10	eqeq2d 2740	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑎 = (1st ‘𝑝) ↔ 𝑎 = (1st ‘⟨𝑎, 𝑏⟩)))
12		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (2nd ‘𝑝) = (2nd ‘⟨𝑎, 𝑏⟩))
13	12	eqeq2d 2740	. . . . . . . . . . . . 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 248	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 → 𝜓))
19	18	adantr 480	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝜑 → 𝜓))
20	19	impcom 407	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵))) → 𝜓)
21	2, 20	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵))) → (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓))
22	21	ex 412	. . . . 5 ⊢ (𝜑 → ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓)))
23	22	2eximdv 1919	. . . 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 476	. . . . 5 ⊢ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓) → 𝜑)
26	25	a1i 11	. . . 4 ⊢ (𝑝 ∈ (𝐴 × 𝐵) → ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓) → 𝜑))
27	26	exlimdvv 1934	. . 3 ⊢ (𝑝 ∈ (𝐴 × 𝐵) → (∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓) → 𝜑))
28	24, 27	impbid 212	. 2 ⊢ (𝑝 ∈ (𝐴 × 𝐵) → (𝜑 ↔ ∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓)))
29	28	reubiia 3350	1 ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃!wreu 3341  ⟨cop 4583   × cxp 5617  ‘cfv 6482  1^st c1st 7922  2^nd c2nd 7923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6438  df-fun 6484  df-fv 6490  df-1st 7924  df-2nd 7925

This theorem is referenced by:  2sqreuopb  27377