Theorem cflecard 4900

Description: Cofinality is bounded by the cardinality of its argument.

Assertion

Ref	Expression
cflecard	|- (cf` A) (_ (card` A)

Proof of Theorem cflecard

Step	Hyp	Ref	Expression
1		cfval 4894	. . 3 |- (A e. On -> (cf` A) = |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
2		ssid 2078	. . . . . . . . . 10 |- A (_ A
3		ssid 2078	. . . . . . . . . . . 12 |- z (_ z
4		sseq2 2081	. . . . . . . . . . . . 13 |- (w = z -> (z (_ w <-> z (_ z))
5	4	rcla4ev 1875	. . . . . . . . . . . 12 |- ((z e. A /\ z (_ z) -> E.w e. A z (_ w)
6	3, 5	mpan2 696	. . . . . . . . . . 11 |- (z e. A -> E.w e. A z (_ w)
7	6	rgen 1697	. . . . . . . . . 10 |- A.z e. A E.w e. A z (_ w
8	2, 7	pm3.2i 285	. . . . . . . . 9 |- (A (_ A /\ A.z e. A E.w e. A z (_ w)
9		fveq2 3722	. . . . . . . . . . . 12 |- (y = A -> (card` y) = (card` A))
10	9	eqeq2d 1485	. . . . . . . . . . 11 |- (y = A -> (x = (card` y) <-> x = (card` A)))
11		sseq1 2080	. . . . . . . . . . . 12 |- (y = A -> (y (_ A <-> A (_ A))
12		rexeq1 1786	. . . . . . . . . . . . 13 |- (y = A -> (E.w e. y z (_ w <-> E.w e. A z (_ w))
13	12	ralbidv 1662	. . . . . . . . . . . 12 |- (y = A -> (A.z e. A E.w e. y z (_ w <-> A.z e. A E.w e. A z (_ w))
14	11, 13	anbi12d 628	. . . . . . . . . . 11 |- (y = A -> ((y (_ A /\ A.z e. A E.w e. y z (_ w) <-> (A (_ A /\ A.z e. A E.w e. A z (_ w)))
15	10, 14	anbi12d 628	. . . . . . . . . 10 |- (y = A -> ((x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) <-> (x = (card` A) /\ (A (_ A /\ A.z e. A E.w e. A z (_ w))))
16	15	cla4egv 1861	. . . . . . . . 9 |- (A e. On -> ((x = (card` A) /\ (A (_ A /\ A.z e. A E.w e. A z (_ w)) -> E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
17	8, 16	mpan2i 699	. . . . . . . 8 |- (A e. On -> (x = (card` A) -> E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
18	17	19.21aiv 1286	. . . . . . 7 |- (A e. On -> A.x(x = (card` A) -> E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
19		ss2ab 2114	. . . . . . 7 |- ({x | x = (card` A)} (_ {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} <-> A.x(x = (card` A) -> E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
20	18, 19	sylibr 200	. . . . . 6 |- (A e. On -> {x | x = (card` A)} (_ {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
21		df-sn 2410	. . . . . 6 |- {(card` A)} = {x | x = (card` A)}
22	20, 21	syl5ss 2103	. . . . 5 |- (A e. On -> {(card` A)} (_ {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
23		intss 2552	. . . . 5 |- ({(card` A)} (_ {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))} (_ |^|{(card` A)})
24	22, 23	syl 10	. . . 4 |- (A e. On -> |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} (_ |^|{(card` A)})
25		fvex 3730	. . . . 5 |- (card` A) e. V
26	25	intsn 2562	. . . 4 |- |^|{(card` A)} = (card` A)
27	24, 26	syl6ss 2105	. . 3 |- (A e. On -> |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} (_ (card` A))
28	1, 27	eqsstrd 2093	. 2 |- (A e. On -> (cf` A) (_ (card` A))
29		0ss 2299	. . 3 |- (/) (_ (card` A)
30		cffnon 4895	. . . . . . . 8 |- cf Fn On
31		fndm 3585	. . . . . . . 8 |- (cf Fn On -> dom cf = On)
32	30, 31	ax-mp 7	. . . . . . 7 |- dom cf = On
33	32	eleq2i 1537	. . . . . 6 |- (A e. dom cf <-> A e. On)
34	33	negbii 187	. . . . 5 |- (-. A e. dom cf <-> -. A e. On)
35		ndmfv 3743	. . . . 5 |- (-. A e. dom cf -> (cf` A) = (/))
36	34, 35	sylbir 201	. . . 4 |- (-. A e. On -> (cf` A) = (/))
37	36	sseq1d 2086	. . 3 |- (-. A e. On -> ((cf` A) (_ (card` A) <-> (/) (_ (card` A)))
38	29, 37	mpbiri 194	. 2 |- (-. A e. On -> (cf` A) (_ (card` A))
39	28, 38	pm2.61i 126	1 |- (cf` A) (_ (card` A)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  A.wral 1644  E.wrex 1645   (_ wss 2045  (/)c0 2278  {csn 2407  |^|cint 2531  Oncon0 2946  dom cdm 3168   Fn wfn 3175  ` cfv 3180  cardccrd 4801  cfccf 4803

This theorem is referenced by:  cfle 4901

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2701  ax-pow 2740  ax-pr 2777  ax-un 2864

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2502  df-int 2532  df-br 2618  df-opab 2665  df-id 2833  df-xp 3182  df-rel 3183  df-cnv 3184  df-co 3185  df-dm 3186  df-rn 3187  df-res 3188  df-ima 3189  df-fun 3190  df-fn 3191  df-fv 3196  df-cf 4806

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