Theorem aceq5lem1 3558

Description: Lemma for aceq5 3563.

Assertion

Ref	Expression
aceq5lem1	⊢ (∃!v v ∈ (({w} × w) ∩ y) ↔ ∃!g(g ∈ w ∧ ⟨w, g⟩ ∈ y))

Distinct variable group(s):   w,v,y,g

Proof of Theorem aceq5lem1

Step	Hyp	Ref	Expression
1		elin 1635	. . . 4 ⊢ (v ∈ (({w} × w) ∩ y) ↔ (v ∈ ({w} × w) ∧ v ∈ y))
2		elxp 2442	. . . . . 6 ⊢ (v ∈ ({w} × w) ↔ ∃t∃g(v = ⟨t, g⟩ ∧ (t ∈ {w} ∧ g ∈ w)))
3		excom 728	. . . . . 6 ⊢ (∃t∃g(v = ⟨t, g⟩ ∧ (t ∈ {w} ∧ g ∈ w)) ↔ ∃g∃t(v = ⟨t, g⟩ ∧ (t ∈ {w} ∧ g ∈ w)))
4	2, 3	bitr 151	. . . . 5 ⊢ (v ∈ ({w} × w) ↔ ∃g∃t(v = ⟨t, g⟩ ∧ (t ∈ {w} ∧ g ∈ w)))
5	4	anbi1i 368	. . . 4 ⊢ ((v ∈ ({w} × w) ∧ v ∈ y) ↔ (∃g∃t(v = ⟨t, g⟩ ∧ (t ∈ {w} ∧ g ∈ w)) ∧ v ∈ y))
6		19.41vv 964	. . . . 5 ⊢ (∃g∃t((v = ⟨t, g⟩ ∧ (t ∈ {w} ∧ g ∈ w)) ∧ v ∈ y) ↔ (∃g∃t(v = ⟨t, g⟩ ∧ (t ∈ {w} ∧ g ∈ w)) ∧ v ∈ y))
7		an23 371	. . . . . . . . 9 ⊢ (((v = ⟨t, g⟩ ∧ (t ∈ {w} ∧ g ∈ w)) ∧ v ∈ y) ↔ ((v = ⟨t, g⟩ ∧ v ∈ y) ∧ (t ∈ {w} ∧ g ∈ w)))
8		eleq1 1149	. . . . . . . . . . 11 ⊢ (v = ⟨t, g⟩ → (v ∈ y ↔ ⟨t, g⟩ ∈ y))
9	8	pm5.32i 489	. . . . . . . . . 10 ⊢ ((v = ⟨t, g⟩ ∧ v ∈ y) ↔ (v = ⟨t, g⟩ ∧ ⟨t, g⟩ ∈ y))
10		elsn 1820	. . . . . . . . . . 11 ⊢ (t ∈ {w} ↔ t = w)
11	10	anbi1i 368	. . . . . . . . . 10 ⊢ ((t ∈ {w} ∧ g ∈ w) ↔ (t = w ∧ g ∈ w))
12	9, 11	anbi12i 369	. . . . . . . . 9 ⊢ (((v = ⟨t, g⟩ ∧ v ∈ y) ∧ (t ∈ {w} ∧ g ∈ w)) ↔ ((v = ⟨t, g⟩ ∧ ⟨t, g⟩ ∈ y) ∧ (t = w ∧ g ∈ w)))
13		an4 388	. . . . . . . . . 10 ⊢ (((v = ⟨t, g⟩ ∧ ⟨t, g⟩ ∈ y) ∧ (t = w ∧ g ∈ w)) ↔ ((v = ⟨t, g⟩ ∧ t = w) ∧ (⟨t, g⟩ ∈ y ∧ g ∈ w)))
14		ancom 333	. . . . . . . . . . 11 ⊢ ((v = ⟨t, g⟩ ∧ t = w) ↔ (t = w ∧ v = ⟨t, g⟩))
15		ancom 333	. . . . . . . . . . 11 ⊢ ((⟨t, g⟩ ∈ y ∧ g ∈ w) ↔ (g ∈ w ∧ ⟨t, g⟩ ∈ y))
16	14, 15	anbi12i 369	. . . . . . . . . 10 ⊢ (((v = ⟨t, g⟩ ∧ t = w) ∧ (⟨t, g⟩ ∈ y ∧ g ∈ w)) ↔ ((t = w ∧ v = ⟨t, g⟩) ∧ (g ∈ w ∧ ⟨t, g⟩ ∈ y)))
17		anass 336	. . . . . . . . . 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	3bitr 155	. . . . . . . . 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	3bitr 155	. . . . . . . 8 ⊢ (((v = ⟨t, g⟩ ∧ (t ∈ {w} ∧ g ∈ w)) ∧ v ∈ y) ↔ (t = w ∧ (v = ⟨t, g⟩ ∧ (g ∈ w ∧ ⟨t, g⟩ ∈ y))))
20	19	biex 733	. . . . . . 7 ⊢ (∃t((v = ⟨t, g⟩ ∧ (t ∈ {w} ∧ g ∈ w)) ∧ v ∈ y) ↔ ∃t(t = w ∧ (v = ⟨t, g⟩ ∧ (g ∈ w ∧ ⟨t, g⟩ ∈ y))))
21		visset 1350	. . . . . . . 8 ⊢ w ∈ V
22		opeq1 1876	. . . . . . . . . 10 ⊢ (t = w → ⟨t, g⟩ = ⟨w, g⟩)
23	22	cleq2d 1112	. . . . . . . . 9 ⊢ (t = w → (v = ⟨t, g⟩ ↔ v = ⟨w, g⟩))
24	22	eleq1d 1155	. . . . . . . . . 10 ⊢ (t = w → (⟨t, g⟩ ∈ y ↔ ⟨w, g⟩ ∈ y))
25	24	anbi2d 468	. . . . . . . . 9 ⊢ (t = w → ((g ∈ w ∧ ⟨t, g⟩ ∈ y) ↔ (g ∈ w ∧ ⟨w, g⟩ ∈ y)))
26	23, 25	anbi12d 476	. . . . . . . 8 ⊢ (t = w → ((v = ⟨t, g⟩ ∧ (g ∈ w ∧ ⟨t, g⟩ ∈ y)) ↔ (v = ⟨w, g⟩ ∧ (g ∈ w ∧ ⟨w, g⟩ ∈ y))))
27	21, 26	ceqsexv 1371	. . . . . . 7 ⊢ (∃t(t = w ∧ (v = ⟨t, g⟩ ∧ (g ∈ w ∧ ⟨t, g⟩ ∈ y))) ↔ (v = ⟨w, g⟩ ∧ (g ∈ w ∧ ⟨w, g⟩ ∈ y)))
28	20, 27	bitr 151	. . . . . 6 ⊢ (∃t((v = ⟨t, g⟩ ∧ (t ∈ {w} ∧ g ∈ w)) ∧ v ∈ y) ↔ (v = ⟨w, g⟩ ∧ (g ∈ w ∧ ⟨w, g⟩ ∈ y)))
29	28	biex 733	. . . . 5 ⊢ (∃g∃t((v = ⟨t, g⟩ ∧ (t ∈ {w} ∧ g ∈ w)) ∧ v ∈ y) ↔ ∃g(v = ⟨w, g⟩ ∧ (g ∈ w ∧ ⟨w, g⟩ ∈ y)))
30	6, 29	bitr3 153	. . . 4 ⊢ ((∃g∃t(v = ⟨t, g⟩ ∧ (t ∈ {w} ∧ g ∈ w)) ∧ v ∈ y) ↔ ∃g(v = ⟨w, g⟩ ∧ (g ∈ w ∧ ⟨w, g⟩ ∈ y)))
31	1, 5, 30	3bitr 155	. . 3 ⊢ (v ∈ (({w} × w) ∩ y) ↔ ∃g(v = ⟨w, g⟩ ∧ (g ∈ w ∧ ⟨w, g⟩ ∈ y)))
32	31	bieu 1014	. 2 ⊢ (∃!v v ∈ (({w} × w) ∩ y) ↔ ∃!v∃g(v = ⟨w, g⟩ ∧ (g ∈ w ∧ ⟨w, g⟩ ∈ y)))
33		euop2 1912	. 2 ⊢ (∃!v∃g(v = ⟨w, g⟩ ∧ (g ∈ w ∧ ⟨w, g⟩ ∈ y)) ↔ ∃!g(g ∈ w ∧ ⟨w, g⟩ ∈ y))
34	32, 33	bitr 151	1 ⊢ (∃!v v ∈ (({w} × w) ∩ y) ↔ ∃!g(g ∈ w ∧ ⟨w, g⟩ ∈ y))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = weq 797   ∈ wel 803  ∃!weu 1007   = wceq 1091   ∈ wcel 1092   ∩ cin 1486  {csn 1808  ⟨cop 1810   × cxp 2408

This theorem is referenced by:  aceq5lem5 3562

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424

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