Theorem opreuopreu 7731

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 5574	. . . 4 ⊢ (𝑝 ∈ (𝐴 × 𝐵) → ∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)))
2		simprl 769	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵))) → 𝑝 = ⟨𝑎, 𝑏⟩)
3		vex 3496	. . . . . . . . . . . . . . 15 ⊢ 𝑎 ∈ V
4		vex 3496	. . . . . . . . . . . . . . 15 ⊢ 𝑏 ∈ V
5	3, 4	op1st 7694	. . . . . . . . . . . . . 14 ⊢ (1st ‘⟨𝑎, 𝑏⟩) = 𝑎
6	5	eqcomi 2829	. . . . . . . . . . . . 13 ⊢ 𝑎 = (1st ‘⟨𝑎, 𝑏⟩)
7	3, 4	op2nd 7695	. . . . . . . . . . . . . 14 ⊢ (2nd ‘⟨𝑎, 𝑏⟩) = 𝑏
8	7	eqcomi 2829	. . . . . . . . . . . . 13 ⊢ 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩)
9	6, 8	pm3.2i 473	. . . . . . . . . . . 12 ⊢ (𝑎 = (1st ‘⟨𝑎, 𝑏⟩) ∧ 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩))
10		fveq2 6667	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (1st ‘𝑝) = (1st ‘⟨𝑎, 𝑏⟩))
11	10	eqeq2d 2831	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑎 = (1st ‘𝑝) ↔ 𝑎 = (1st ‘⟨𝑎, 𝑏⟩)))
12		fveq2 6667	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (2nd ‘𝑝) = (2nd ‘⟨𝑎, 𝑏⟩))
13	12	eqeq2d 2831	. . . . . . . . . . . . 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 260	. . . . . . . . . . 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 250	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 → 𝜓))
19	18	adantr 483	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝜑 → 𝜓))
20	19	impcom 410	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵))) → 𝜓)
21	2, 20	jca 514	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵))) → (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓))
22	21	ex 415	. . . . 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 479	. . . . 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 214	. 2 ⊢ (𝑝 ∈ (𝐴 × 𝐵) → (𝜑 ↔ ∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓)))
29	28	reubiia 3389	1 ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536  ∃wex 1779   ∈ wcel 2113  ∃!wreu 3139  ⟨cop 4570   × cxp 5550  ‘cfv 6352  1^st c1st 7684  2^nd c2nd 7685

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5327  ax-un 7458

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-reu 3144  df-rab 3146  df-v 3495  df-sbc 3771  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4465  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4836  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5457  df-xp 5558  df-rel 5559  df-cnv 5560  df-co 5561  df-dm 5562  df-rn 5563  df-iota 6311  df-fun 6354  df-fv 6360  df-1st 7686  df-2nd 7687

This theorem is referenced by:  2sqreuopb  26042