Theorem opreuopreu 7734

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 5577	. . . 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 3497	. . . . . . . . . . . . . . 15 ⊢ 𝑎 ∈ V
4		vex 3497	. . . . . . . . . . . . . . 15 ⊢ 𝑏 ∈ V
5	3, 4	op1st 7697	. . . . . . . . . . . . . 14 ⊢ (1st ‘⟨𝑎, 𝑏⟩) = 𝑎
6	5	eqcomi 2830	. . . . . . . . . . . . 13 ⊢ 𝑎 = (1st ‘⟨𝑎, 𝑏⟩)
7	3, 4	op2nd 7698	. . . . . . . . . . . . . 14 ⊢ (2nd ‘⟨𝑎, 𝑏⟩) = 𝑏
8	7	eqcomi 2830	. . . . . . . . . . . . 13 ⊢ 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩)
9	6, 8	pm3.2i 473	. . . . . . . . . . . 12 ⊢ (𝑎 = (1st ‘⟨𝑎, 𝑏⟩) ∧ 𝑏 = (2nd ‘⟨𝑎, 𝑏⟩))
10		fveq2 6670	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (1st ‘𝑝) = (1st ‘⟨𝑎, 𝑏⟩))
11	10	eqeq2d 2832	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑎 = (1st ‘𝑝) ↔ 𝑎 = (1st ‘⟨𝑎, 𝑏⟩)))
12		fveq2 6670	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (2nd ‘𝑝) = (2nd ‘⟨𝑎, 𝑏⟩))
13	12	eqeq2d 2832	. . . . . . . . . . . . 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 1920	. . . 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 1935	. . 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 3390	1 ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∃!wreu 3140  ⟨cop 4573   × cxp 5553  ‘cfv 6355  1^st c1st 7687  2^nd c2nd 7688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-iota 6314  df-fun 6357  df-fv 6363  df-1st 7689  df-2nd 7690

This theorem is referenced by:  2sqreuopb  26044