Theorem dfac5lem1 10095

Description: Lemma for dfac5 10100. (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 3923	. . . 4 ⊢ (𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦) ↔ (𝑣 ∈ ({𝑤} × 𝑤) ∧ 𝑣 ∈ 𝑦))
2		elxp 5674	. . . . . 6 ⊢ (𝑣 ∈ ({𝑤} × 𝑤) ↔ ∃𝑡∃𝑔(𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)))
3		excom 2199	. . . . . 6 ⊢ (∃𝑡∃𝑔(𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)) ↔ ∃𝑔∃𝑡(𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)))
4	2, 3	bitri 278	. . . . 5 ⊢ (𝑣 ∈ ({𝑤} × 𝑤) ↔ ∃𝑔∃𝑡(𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)))
5	4	anbi1i 635	. . . 4 ⊢ ((𝑣 ∈ ({𝑤} × 𝑤) ∧ 𝑣 ∈ 𝑦) ↔ (∃𝑔∃𝑡(𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)) ∧ 𝑣 ∈ 𝑦))
6		19.41vv 1973	. . . . 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 658	. . . . . . . . 9 ⊢ (((𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)) ∧ 𝑣 ∈ 𝑦) ↔ ((𝑣 = ⟨𝑡, 𝑔⟩ ∧ 𝑣 ∈ 𝑦) ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)))
8		eleq1 2853	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨𝑡, 𝑔⟩ → (𝑣 ∈ 𝑦 ↔ ⟨𝑡, 𝑔⟩ ∈ 𝑦))
9	8	pm5.32i 584	. . . . . . . . . 10 ⊢ ((𝑣 = ⟨𝑡, 𝑔⟩ ∧ 𝑣 ∈ 𝑦) ↔ (𝑣 = ⟨𝑡, 𝑔⟩ ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦))
10		velsn 4601	. . . . . . . . . . 11 ⊢ (𝑡 ∈ {𝑤} ↔ 𝑡 = 𝑤)
11	10	anbi1i 635	. . . . . . . . . 10 ⊢ ((𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤) ↔ (𝑡 = 𝑤 ∧ 𝑔 ∈ 𝑤))
12	9, 11	anbi12i 639	. . . . . . . . 9 ⊢ (((𝑣 = ⟨𝑡, 𝑔⟩ ∧ 𝑣 ∈ 𝑦) ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)) ↔ ((𝑣 = ⟨𝑡, 𝑔⟩ ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦) ∧ (𝑡 = 𝑤 ∧ 𝑔 ∈ 𝑤)))
13		an4 668	. . . . . . . . . 10 ⊢ (((𝑣 = ⟨𝑡, 𝑔⟩ ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦) ∧ (𝑡 = 𝑤 ∧ 𝑔 ∈ 𝑤)) ↔ ((𝑣 = ⟨𝑡, 𝑔⟩ ∧ 𝑡 = 𝑤) ∧ (⟨𝑡, 𝑔⟩ ∈ 𝑦 ∧ 𝑔 ∈ 𝑤)))
14		ancom 465	. . . . . . . . . . 11 ⊢ ((𝑣 = ⟨𝑡, 𝑔⟩ ∧ 𝑡 = 𝑤) ↔ (𝑡 = 𝑤 ∧ 𝑣 = ⟨𝑡, 𝑔⟩))
15		ancom 465	. . . . . . . . . . 11 ⊢ ((⟨𝑡, 𝑔⟩ ∈ 𝑦 ∧ 𝑔 ∈ 𝑤) ↔ (𝑔 ∈ 𝑤 ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦))
16	14, 15	anbi12i 639	. . . . . . . . . 10 ⊢ (((𝑣 = ⟨𝑡, 𝑔⟩ ∧ 𝑡 = 𝑤) ∧ (⟨𝑡, 𝑔⟩ ∈ 𝑦 ∧ 𝑔 ∈ 𝑤)) ↔ ((𝑡 = 𝑤 ∧ 𝑣 = ⟨𝑡, 𝑔⟩) ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦)))
17		anass 473	. . . . . . . . . 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 300	. . . . . . . . 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 300	. . . . . . . 8 ⊢ (((𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)) ∧ 𝑣 ∈ 𝑦) ↔ (𝑡 = 𝑤 ∧ (𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦))))
20	19	exbii 1871	. . . . . . 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 2776	. . . . . . . . 9 ⊢ (𝑡 = 𝑤 → (𝑣 = ⟨𝑡, 𝑔⟩ ↔ 𝑣 = ⟨𝑤, 𝑔⟩))
23	21	eleq1d 2850	. . . . . . . . . 10 ⊢ (𝑡 = 𝑤 → (⟨𝑡, 𝑔⟩ ∈ 𝑦 ↔ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
24	23	anbi2d 641	. . . . . . . . 9 ⊢ (𝑡 = 𝑤 → ((𝑔 ∈ 𝑤 ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦) ↔ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
25	22, 24	anbi12d 643	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → ((𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦)) ↔ (𝑣 = ⟨𝑤, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))))
26	25	equsexvw 2028	. . . . . . 7 ⊢ (∃𝑡(𝑡 = 𝑤 ∧ (𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑡, 𝑔⟩ ∈ 𝑦))) ↔ (𝑣 = ⟨𝑤, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
27	20, 26	bitri 278	. . . . . 6 ⊢ (∃𝑡((𝑣 = ⟨𝑡, 𝑔⟩ ∧ (𝑡 ∈ {𝑤} ∧ 𝑔 ∈ 𝑤)) ∧ 𝑣 ∈ 𝑦) ↔ (𝑣 = ⟨𝑤, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
28	27	exbii 1871	. . . . 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 280	. . . 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 300	. . 3 ⊢ (𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦) ↔ ∃𝑔(𝑣 = ⟨𝑤, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
31	30	eubii 2615	. 2 ⊢ (∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦) ↔ ∃!𝑣∃𝑔(𝑣 = ⟨𝑤, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
32		vex 3461	. . 3 ⊢ 𝑤 ∈ V
33	32	euop2 5485	. 2 ⊢ (∃!𝑣∃𝑔(𝑣 = ⟨𝑤, 𝑔⟩ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)) ↔ ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
34	31, 33	bitri 278	1 ⊢ (∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦) ↔ ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∃!weu 2598   ∩ cin 3906  {csn 4585  ⟨cop 4591   × cxp 5649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-opab 5167  df-xp 5657

This theorem is referenced by:  dfac5lem5  10099