Theorem cflim 3704

Description: Value of the cofinality function at a limit ordinal. Part of Definition of cofinality of [Enderton] p. 257.

Assertion

Ref	Expression
cflim	⊢ ((A ∈ B ∧ Lim A) → (cf ‘A) = ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))})

Distinct variable group(s):   x,y,A

Proof of Theorem cflim

Step	Hyp	Ref	Expression
1		limsuc 2361	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Lim A → (v ∈ A ↔ suc v ∈ A))
2	1	biimpd 135	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim A → (v ∈ A → suc v ∈ A))
3		sseq1 1521	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (z = suc v → (z ⊆ w ↔ suc v ⊆ w))
4	3	birexdv 1220	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (z = suc v → (∃w ∈ y z ⊆ w ↔ ∃w ∈ y suc v ⊆ w))
5	4	rcla4v 1402	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀z ∈ A ∃w ∈ y z ⊆ w → (suc v ∈ A → ∃w ∈ y suc v ⊆ w))
6		visset 1350	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ v ∈ V
7		sucssel 2321	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (v ∈ V → (suc v ⊆ w → v ∈ w))
8	6, 7	ax-mp 6	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (suc v ⊆ w → v ∈ w)
9	8	r19.22si 1275	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃w ∈ y suc v ⊆ w → ∃w ∈ y v ∈ w)
10		eluni2 1923	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (v ∈ ∪y ↔ ∃w ∈ y v ∈ w)
11	9, 10	sylibr 175	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃w ∈ y suc v ⊆ w → v ∈ ∪y)
12	5, 11	syl6 23	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀z ∈ A ∃w ∈ y z ⊆ w → (suc v ∈ A → v ∈ ∪y))
13	2, 12	syl9 55	. . . . . . . . . . . . . . . . . 18 ⊢ (Lim A → (∀z ∈ A ∃w ∈ y z ⊆ w → (v ∈ A → v ∈ ∪y)))
14	13	r19.21adv 1262	. . . . . . . . . . . . . . . . 17 ⊢ (Lim A → (∀z ∈ A ∃w ∈ y z ⊆ w → ∀v ∈ A v ∈ ∪y))
15		dfss3 1498	. . . . . . . . . . . . . . . . 17 ⊢ (A ⊆ ∪y ↔ ∀v ∈ A v ∈ ∪y)
16	14, 15	syl6ibr 186	. . . . . . . . . . . . . . . 16 ⊢ (Lim A → (∀z ∈ A ∃w ∈ y z ⊆ w → A ⊆ ∪y))
17	16	adantr 306	. . . . . . . . . . . . . . 15 ⊢ ((Lim A ∧ y ⊆ A) → (∀z ∈ A ∃w ∈ y z ⊆ w → A ⊆ ∪y))
18		limuni 2284	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim A → A = ∪A)
19	18	sseq2d 1528	. . . . . . . . . . . . . . . . . 18 ⊢ (Lim A → (∪y ⊆ A ↔ ∪y ⊆ ∪A))
20		uniss 1936	. . . . . . . . . . . . . . . . . 18 ⊢ (y ⊆ A → ∪y ⊆ ∪A)
21	19, 20	syl5bir 184	. . . . . . . . . . . . . . . . 17 ⊢ (Lim A → (y ⊆ A → ∪y ⊆ A))
22	21	imp 277	. . . . . . . . . . . . . . . 16 ⊢ ((Lim A ∧ y ⊆ A) → ∪y ⊆ A)
23	22	a1d 14	. . . . . . . . . . . . . . 15 ⊢ ((Lim A ∧ y ⊆ A) → (∀z ∈ A ∃w ∈ y z ⊆ w → ∪y ⊆ A))
24	17, 23	jcad 455	. . . . . . . . . . . . . 14 ⊢ ((Lim A ∧ y ⊆ A) → (∀z ∈ A ∃w ∈ y z ⊆ w → (A ⊆ ∪y ∧ ∪y ⊆ A)))
25		eqss 1516	. . . . . . . . . . . . . 14 ⊢ (A = ∪y ↔ (A ⊆ ∪y ∧ ∪y ⊆ A))
26	24, 25	syl6ibr 186	. . . . . . . . . . . . 13 ⊢ ((Lim A ∧ y ⊆ A) → (∀z ∈ A ∃w ∈ y z ⊆ w → A = ∪y))
27	26	exp 291	. . . . . . . . . . . 12 ⊢ (Lim A → (y ⊆ A → (∀z ∈ A ∃w ∈ y z ⊆ w → A = ∪y)))
28	27	imdistand 342	. . . . . . . . . . 11 ⊢ (Lim A → ((y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w) → (y ⊆ A ∧ A = ∪y)))
29	28	anim2d 433	. . . . . . . . . 10 ⊢ (Lim A → ((x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w)) → (x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))))
30	29	19.22dv 947	. . . . . . . . 9 ⊢ (Lim A → (∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w)) → ∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))))
31	30	19.21aiv 943	. . . . . . . 8 ⊢ (Lim A → ∀x(∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w)) → ∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))))
32		ss2ab 1551	. . . . . . . 8 ⊢ ({x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))} ⊆ {x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))} ↔ ∀x(∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w)) → ∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))))
33	31, 32	sylibr 175	. . . . . . 7 ⊢ (Lim A → {x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))} ⊆ {x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))})
34		intss 1983	. . . . . . 7 ⊢ ({x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))} ⊆ {x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))} → ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))} ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))})
35	33, 34	syl 12	. . . . . 6 ⊢ (Lim A → ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))} ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))})
36	35	adantl 305	. . . . 5 ⊢ ((A ∈ V ∧ Lim A) → ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))} ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))})
37		limelon 2286	. . . . . 6 ⊢ ((A ∈ V ∧ Lim A) → A ∈ On)
38		cfval 3701	. . . . . 6 ⊢ (A ∈ On → (cf ‘A) = ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))})
39	37, 38	syl 12	. . . . 5 ⊢ ((A ∈ V ∧ Lim A) → (cf ‘A) = ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))})
40	36, 39	sseqtr4d 1537	. . . 4 ⊢ ((A ∈ V ∧ Lim A) → ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))} ⊆ (cf ‘A))
41		cfub 3703	. . . . 5 ⊢ (cf ‘A) ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y))}
42		eqimss 1548	. . . . . . . . . 10 ⊢ (A = ∪y → A ⊆ ∪y)
43	42	anim2i 270<­TD>	. . . . . . . . 9 ⊢ ((y ⊆ A ∧ A = ∪y) → (y ⊆ A ∧ A ⊆ ∪y))
44	43	anim2i 270	. . . . . . . 8 ⊢ ((x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y)) → (x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y)))
45	44	19.22i 723	. . . . . . 7 ⊢ (∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y)) → ∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y)))
46	45	ss2abi 1552	. . . . . 6 ⊢ {x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))} ⊆ {x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y))}
47		intss 1983	. . . . . 6 ⊢ ({x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))} ⊆ {x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y))} → ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y))} ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))})
48	46, 47	ax-mp 6	. . . . 5 ⊢ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y))} ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))}
49	41, 48	sstri 1512	. . . 4 ⊢ (cf ‘A) ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))}
50	40, 49	jctil 240	. . 3 ⊢ ((A ∈ V ∧ Lim A) → ((cf ‘A) ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))} ∧ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))} ⊆ (cf ‘A)))
51		eqss 1516	. . 3 ⊢ ((cf ‘A) = ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))} ↔ ((cf ‘A) ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))} ∧ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))} ⊆ (cf ‘A)))
52	50, 51	sylibr 175	. 2 ⊢ ((A ∈ V ∧ Lim A) → (cf ‘A) = ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))})
53		elisset 1354	. 2 ⊢ (A ∈ B → A ∈ V)
54	52, 53	sylan 343	1 ⊢ ((A ∈ B ∧ Lim A) → (cf ‘A) = ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))})

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348   ⊆ wss 1487  ∪cuni 1919  ∩cint 1965  Oncon0 2199  Lim wlim 2200  suc csuc 2201   ‘cfv 2422  cardccrd 3620  cfccf 3622

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-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-cf 3625

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