Theorem cflecard 6060

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

Assertion

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

Proof of Theorem cflecard

Step	Hyp	Ref	Expression
1		cfval 6054	. . 3 |- (A e. On -> (cf` A) = |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))})
2		ssid 2634	. . . . . . . . 9 |- A C_ A
3		ssid 2634	. . . . . . . . . . 11 |- z C_ z
4		sseq2 2639	. . . . . . . . . . . 12 |- (w = z -> (z C_ w <-> z C_ z))
5	4	rcla4ev 2381	. . . . . . . . . . 11 |- ((z e. A /\ z C_ z) -> E.w e. A z C_ w)
6	3, 5	mpan2 760	. . . . . . . . . 10 |- (z e. A -> E.w e. A z C_ w)
7	6	rgen 2159	. . . . . . . . 9 |- A.z e. A E.w e. A z C_ w
8	2, 7	pm3.2i 307	. . . . . . . 8 |- (A C_ A /\ A.z e. A E.w e. A z C_ w)
9		fveq2 4681	. . . . . . . . . . 11 |- (y = A -> (card` y) = (card` A))
10	9	eqeq2d 1895	. . . . . . . . . 10 |- (y = A -> (x = (card` y) <-> x = (card` A)))
11		sseq1 2637	. . . . . . . . . . 11 |- (y = A -> (y C_ A <-> A C_ A))
12		rexeq 2267	. . . . . . . . . . . 12 |- (y = A -> (E.w e. y z C_ w <-> E.w e. A z C_ w))
13	12	ralbidv 2123	. . . . . . . . . . 11 |- (y = A -> (A.z e. A E.w e. y z C_ w <-> A.z e. A E.w e. A z C_ w))
14	11, 13	anbi12d 690	. . . . . . . . . 10 |- (y = A -> ((y C_ A /\ A.z e. A E.w e. y z C_ w) <-> (A C_ A /\ A.z e. A E.w e. A z C_ w)))
15	10, 14	anbi12d 690	. . . . . . . . 9 |- (y = A -> ((x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w)) <-> (x = (card` A) /\ (A C_ A /\ A.z e. A E.w e. A z C_ w))))
16	15	cla4egv 2365	. . . . . . . 8 |- (A e. On -> ((x = (card` A) /\ (A C_ A /\ A.z e. A E.w e. A z C_ w)) -> E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))))
17	8, 16	mpan2i 763	. . . . . . 7 |- (A e. On -> (x = (card` A) -> E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))))
18	17	ss2abdv 2680	. . . . . 6 |- (A e. On -> {x | x = (card` A)} C_ {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))})
19		df-sn 3049	. . . . . 6 |- {(card` A)} = {x | x = (card` A)}
20	18, 19	syl5ss 2661	. . . . 5 |- (A e. On -> {(card` A)} C_ {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))})
21		intss 3239	. . . . 5 |- ({(card` A)} C_ {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} -> |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} C_ |^|{(card` A)})
22	20, 21	syl 12	. . . 4 |- (A e. On -> |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} C_ |^|{(card` A)})
23		fvex 4689	. . . . 5 |- (card` A) e. _V
24	23	intsn 3252	. . . 4 |- |^|{(card` A)} = (card` A)
25	22, 24	syl6ss 2663	. . 3 |- (A e. On -> |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} C_ (card` A))
26	1, 25	eqsstrd 2651	. 2 |- (A e. On -> (cf` A) C_ (card` A))
27		0ss 2900	. . 3 |- (/) C_ (card` A)
28		cffnon 6055	. . . . . . . 8 |- cf Fn On
29		fndm 4512	. . . . . . . 8 |- (cf Fn On -> dom cf = On)
30	28, 29	ax-mp 7	. . . . . . 7 |- dom cf = On
31	30	eleq2i 1961	. . . . . 6 |- (A e. dom cf <-> A e. On)
32	31	notbii 204	. . . . 5 |- (-. A e. dom cf <-> -. A e. On)
33		ndmfv 4702	. . . . 5 |- (-. A e. dom cf -> (cf` A) = (/))
34	32, 33	sylbir 218	. . . 4 |- (-. A e. On -> (cf` A) = (/))
35	34	sseq1d 2644	. . 3 |- (-. A e. On -> ((cf` A) C_ (card` A) <-> (/) C_ (card` A)))
36	27, 35	mpbiri 211	. 2 |- (-. A e. On -> (cf` A) C_ (card` A))
37	26, 36	pm2.61i 140	1 |- (cf` A) C_ (card` A)

Syntax hints:  -. wn 2   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  A.wral 2105  E.wrex 2106   C_ wss 2593  (/)c0 2875  {csn 3044  |^|cint 3214  Oncon0 3657  dom cdm 3986   Fn wfn 3993  ` cfv 3998  cardccrd 5859  cfccf 5861

This theorem is referenced by:  cfle 6061

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-cf 5864

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