Theorem cardcf 3706

Description: Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103.

Assertion

Ref	Expression
cardcf	|- (card` (cf` A)) = (cf` A)

Proof of Theorem cardcf

Step	Hyp	Ref	Expression
1		cfval 3701	. . . 4 |- (A e. On -> (cf` A) = |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
2		fvex 2838	. . . . . . 7 |- (cf` A) e. V
3	1	eleq1d 1155	. . . . . . 7 |- (A e. On -> ((cf` A) e. V <-> |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} e. V))
4	2, 3	mpbii 168	. . . . . 6 |- (A e. On -> |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} e. V)
5		intex 1986	. . . . . 6 |- (-. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} = (/) <-> |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} e. V)
6	4, 5	sylibr 175	. . . . 5 |- (A e. On -> -. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} = (/))
7		cardon 3634	. . . . . . . 8 |- (card` v) e. On
8		visset 1350	. . . . . . . . . . 11 |- v e. V
9		cleq1 1107	. . . . . . . . . . . . 13 |- (x = v -> (x = (card` y) <-> v = (card` y)))
10	9	anbi1d 469	. . . . . . . . . . . 12 |- (x = v -> ((x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) <-> (v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
11	10	biexdv 936	. . . . . . . . . . 11 |- (x = v -> (E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) <-> E.y(v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
12	8, 11	elab 1415	. . . . . . . . . 10 |- (v e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} <-> E.y(v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)))
13		fveq2 2832	. . . . . . . . . . . . . 14 |- (v = (card` y) -> (card` v) = (card` (card` y)))
14		cardcard 3655	. . . . . . . . . . . . . 14 |- (card` (card` y)) = (card` y)
15	13, 14	syl6eq 1140	. . . . . . . . . . . . 13 |- (v = (card` y) -> (card` v) = (card` y))
16		cleq2 1110	. . . . . . . . . . . . 13 |- (v = (card` y) -> ((card` v) = v <-> (card` v) = (card` y)))
17	15, 16	mpbird 171	. . . . . . . . . . . 12 |- (v = (card` y) -> (card` v) = v)
18	17	adantr 306	. . . . . . . . . . 11 |- ((v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) -> (card` v) = v)
19	18	19.23aiv 952	. . . . . . . . . 10 |- (E.y(v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) -> (card` v) = v)
20	12, 19	sylbi 174	. . . . . . . . 9 |- (v e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} -> (card` v) = v)
21	20	eleq1d 1155	. . . . . . . 8 |- (v e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} -> ((card` v) e. On <-> v e. On))
22	7, 21	mpbii 168	. . . . . . 7 |- (v e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} -> v e. On)
23	22	ssriv 1508	. . . . . 6 |- {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} (_ On
24		onint 2261	. . . . . 6 |- (({x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} (_ On /\ -. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} = (/)) -> |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
25	23, 24	mpan 518	. . . . 5 |- (-. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} = (/) -> |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
26	6, 25	syl 12	. . . 4 |- (A e. On -> |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
27	1, 26	eqeltrd 1163	. . 3 |- (A e. On -> (cf` A) e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
28		fveq2 2832	. . . . 5 |- (v = (cf` A) -> (card` v) = (card` (cf` A)))
29		id 9	. . . . 5 |- (v = (cf` A) -> v = (cf` A))
30	28, 29	cleq12d 1115	. . . 4 |- (v = (cf` A) -> ((card` v) = v <-> (card` (cf` A)) = (cf` A)))
31	30, 20	vtoclga 1387	. . 3 |- ((cf` A) e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} -> (card` (cf` A)) = (cf` A))
32	27, 31	syl 12	. 2 |- (A e. On -> (card` (cf` A)) = (cf` A))
33		cffnon 3702	. . . . . . . 8 |- cf Fn On
34		fndm 2723	. . . . . . . 8 |- (cf Fn On -> dom cf = On)
35	33, 34	ax-mp 6	. . . . . . 7 |- dom cf = On
36	35	eleq2i 1153	. . . . . 6 |- (A e. dom cf <-> A e. On)
37	36	negbii 162	. . . . 5 |- (-. A e. dom cf <-> -. A e. On)
38	37	bicomi 150	. . . 4 |- (-. A e. On <-> -. A e. dom cf)
39		ndmfv 2848	. . . 4 |- (-. A e. dom cf -> (cf` A) = (/))
40	38, 39	sylbi 174	. . 3 |- (-. A e. On -> (cf` A) = (/))
41		card0 3630	. . . 4 |- (card` (/)) = (/)
42		fveq2 2832	. . . . 5 |- ((cf` A) = (/) -> (card` (cf` A)) = (card` (/)))
43		id 9	. . . . 5 |- ((cf` A) = (/) -> (cf` A) = (/))
44	42, 43	cleq12d 1115	. . . 4 |- ((cf` A) = (/) -> ((card` (cf` A)) = (cf` A) <-> (card` (/)) = (/)))
45	41, 44	mpbiri 169	. . 3 |- ((cf` A) = (/) -> (card` (cf` A)) = (cf` A))
46	40, 45	syl 12	. 2 |- (-. A e. On -> (card` (cf` A)) = (cf` A))
47	32, 46	pm2.61i 110	1 |- (card` (cf` A)) = (cf` A)

Syntax hints:  -. wn 1   /\ wa 196  E.wex 678   = weq 797  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202  Vcvv 1348   (_ wss 1487  (/)c0 1707  |^|cint 1965  Oncon0 2199  dom cdm 2410   Fn wfn 2417  ` 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  ax-reg 1078  ax-ac 1080

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-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  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-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-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-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-er 3200  df-en 3274  df-card 3623  df-cf 3625

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