Theorem dfac5lem1 10081

Description: Lemma for dfac5 10087. (Contributed by NM, 12-Apr-2004.)

Assertion

Ref	Expression
dfac5lem1	⊢ (∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦) ↔ ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))

Distinct variable group:   𝑤,𝑣,𝑦,𝑔

Proof of Theorem dfac5lem1

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3922	. . . 4 ⊢ (𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦) ↔ (𝑣 ∈ ({𝑤} × 𝑤) ∧ 𝑣 ∈ 𝑦))
2		elxp 5672	. . . . . 6 ⊢ (𝑣 ∈ ({𝑤} × 𝑤) ↔ ∃𝑡∃𝑔(𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)))
3		excom 2198	. . . . . 6 ⊢ (∃𝑡∃𝑔(𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)) ↔ ∃𝑔∃𝑡(𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)))
4	2, 3	bitri 277	. . . . 5 ⊢ (𝑣 ∈ ({𝑤} × 𝑤) ↔ ∃𝑔∃𝑡(𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)))
5	4	anbi1i 633	. . . 4 ⊢ ((𝑣 ∈ ({𝑤} × 𝑤) ∧ 𝑣 ∈ 𝑦) ↔ (∃𝑔∃𝑡(𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)) ∧ 𝑣 ∈ 𝑦))
6		19.41vv 1972	. . . . 5 ⊢ (∃𝑔∃𝑡((𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)) ∧ 𝑣 ∈ 𝑦) ↔ (∃𝑔∃𝑡(𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)) ∧ 𝑣 ∈ 𝑦))
7		an32 656	. . . . . . . . 9 ⊢ (((𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)) ∧ 𝑣 ∈ 𝑦) ↔ ((𝑣 = ⟨𝑡, 𝑔⟩ ∧ 𝑣 ∈ 𝑦) ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)))
8		eleq1 2852	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨𝑡, 𝑔⟩ → (𝑣 ∈ 𝑦 ↔ ⟨𝑡, 𝑔⟩ ∈ 𝑦))
9	8	pm5.32i 582	. . . . . . . . . 10 ⊢ ((𝑣 = ⟨𝑡, 𝑔⟩ ∧ 𝑣 ∈ 𝑦) ↔ (𝑣 = ⟨𝑡, 𝑔⟩ ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦))
10		velsn 4600	. . . . . . . . . . 11 ⊢ (𝑡 ∈ {𝑤} ↔ 𝑡 = 𝑤)
11	10	anbi1i 633	. . . . . . . . . 10 ⊢ ((𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤) ↔ (𝑡 = 𝑤 ∧ 𝑔 ∈ 𝑤))
12	9, 11	anbi12i 637	. . . . . . . . 9 ⊢ (((𝑣 = ⟨𝑡, 𝑔⟩ ∧ 𝑣 ∈ 𝑦) ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)) ↔ ((𝑣 = ⟨𝑡, 𝑔⟩ ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦) ∧ (𝑡 = 𝑤 ∧ 𝑔 ∈ 𝑤)))
13		an4 666	. . . . . . . . . 10 ⊢ (((𝑣 = ⟨𝑡, 𝑔⟩ ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦) ∧ (𝑡 = 𝑤 ∧ 𝑔 ∈ 𝑤)) ↔ ((𝑣 = ⟨𝑡, 𝑔⟩ ∧ 𝑡 = 𝑤) ∧ (⟨𝑡, 𝑔⟩ ∈ 𝑦 ∧ 𝑔 ∈ 𝑤)))
14		ancom 464	. . . . . . . . . . 11 ⊢ ((𝑣 = ⟨𝑡, 𝑔⟩ ∧ 𝑡 = 𝑤) ↔ (𝑡 = 𝑤 ∧ 𝑣 = ⟨𝑡, 𝑔⟩))
15		ancom 464	. . . . . . . . . . 11 ⊢ ((⟨𝑡, 𝑔⟩ ∈ 𝑦 ∧ 𝑔 ∈ 𝑤) ↔ (𝑔 ∈ 𝑤 ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦))
16	14, 15	anbi12i 637	. . . . . . . . . 10 ⊢ (((𝑣 = ⟨𝑡, 𝑔⟩ ∧ 𝑡 = 𝑤) ∧ (⟨𝑡, 𝑔⟩ ∈ 𝑦 ∧ 𝑔 ∈ 𝑤)) ↔ ((𝑡 = 𝑤 ∧ 𝑣 = ⟨𝑡, 𝑔⟩) ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦)))
17		anass 472	. . . . . . . . . 10 ⊢ (((𝑡 = 𝑤 ∧ 𝑣 = ⟨𝑡, 𝑔⟩) ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦)) ↔ (𝑡 = 𝑤 ∧ (𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦))))
18	13, 16, 17	3bitri 299	. . . . . . . . 9 ⊢ (((𝑣 = ⟨𝑡, 𝑔⟩ ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦) ∧ (𝑡 = 𝑤 ∧ 𝑔 ∈ 𝑤)) ↔ (𝑡 = 𝑤 ∧ (𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦))))
19	7, 12, 18	3bitri 299	. . . . . . . 8 ⊢ (((𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)) ∧ 𝑣 ∈ 𝑦) ↔ (𝑡 = 𝑤 ∧ (𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦))))
20	19	exbii 1870	. . . . . . 7 ⊢ (∃𝑡((𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)) ∧ 𝑣 ∈ 𝑦) ↔ ∃𝑡(𝑡 = 𝑤 ∧ (𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦))))
21		opeq1 4833	. . . . . . . . . 10 ⊢ (𝑡 = 𝑤 → ⟨𝑡, 𝑔⟩ = ⟨𝑤, 𝑔⟩)
22	21	eqeq2d 2775	. . . . . . . . 9 ⊢ (𝑡 = 𝑤 → (𝑣 = ⟨𝑡, 𝑔⟩ ↔ 𝑣 = ⟨𝑤, 𝑔⟩))
23	21	eleq1d 2849	. . . . . . . . . 10 ⊢ (𝑡 = 𝑤 → (⟨𝑡, 𝑔⟩ ∈ 𝑦 ↔ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
24	23	anbi2d 639	. . . . . . . . 9 ⊢ (𝑡 = 𝑤 → ((𝑔 ∈ 𝑤 ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦) ↔ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
25	22, 24	anbi12d 641	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → ((𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦)) ↔ (𝑣 = ⟨𝑤, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))))
26	25	equsexvw 2027	. . . . . . 7 ⊢ (∃𝑡(𝑡 = 𝑤 ∧ (𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦))) ↔ (𝑣 = ⟨𝑤, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
27	20, 26	bitri 277	. . . . . 6 ⊢ (∃𝑡((𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)) ∧ 𝑣 ∈ 𝑦) ↔ (𝑣 = ⟨𝑤, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
28	27	exbii 1870	. . . . 5 ⊢ (∃𝑔∃𝑡((𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)) ∧ 𝑣 ∈ 𝑦) ↔ ∃𝑔(𝑣 = ⟨𝑤, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
29	6, 28	bitr3i 279	. . . 4 ⊢ ((∃𝑔∃𝑡(𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)) ∧ 𝑣 ∈ 𝑦) ↔ ∃𝑔(𝑣 = ⟨𝑤, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
30	1, 5, 29	3bitri 299	. . 3 ⊢ (𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦) ↔ ∃𝑔(𝑣 = ⟨𝑤, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
31	30	eubii 2614	. 2 ⊢ (∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦) ↔ ∃!𝑣∃𝑔(𝑣 = ⟨𝑤, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
32		vex 3460	. . 3 ⊢ 𝑤 ∈ V
33	32	euop2 5483	. 2 ⊢ (∃!𝑣∃𝑔(𝑣 = ⟨𝑤, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)) ↔ ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
34	31, 33	bitri 277	1 ⊢ (∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦) ↔ ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1562  ∃wex 1801   ∈ wcel 2144  ∃!weu 2597   ∩ cin 3905  {csn 4584  ⟨cop 4590   × cxp 5647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5165  df-xp 5655

This theorem is referenced by:  dfac5lem5  10085