Theorem cflecard 3707

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 3701	. . 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 1519	. . . . . . . . . 10 |- A (_ A
3		ssid 1519	. . . . . . . . . . . 12 |- z (_ z
4		sseq2 1522	. . . . . . . . . . . . 13 |- (w = z -> (z (_ w <-> z (_ z))
5	4	rcla4ev 1403	. . . . . . . . . . . 12 |- ((z e. A /\ z (_ z) -> E.w e. A z (_ w)
6	3, 5	mpan2 519	. . . . . . . . . . 11 |- (z e. A -> E.w e. A z (_ w)
7	6	rgen 1247	. . . . . . . . . 10 |- A.z e. A E.w e. A z (_ w
8	2, 7	pm3.2i 234	. . . . . . . . 9 |- (A (_ A /\ A.z e. A E.w e. A z (_ w)
9		fveq2 2832	. . . . . . . . . . . 12 |- (y = A -> (card` y) = (card` A))
10	9	cleq2d 1112	. . . . . . . . . . 11 |- (y = A -> (x = (card` y) <-> x = (card` A)))
11		sseq1 1521	. . . . . . . . . . . 12 |- (y = A -> (y (_ A <-> A (_ A))
12		rexeq 1325	. . . . . . . . . . . . 13 |- (y = A -> (E.w e. y z (_ w <-> E.w e. A z (_ w))
13	12	biraldv 1219	. . . . . . . . . . . 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 476	. . . . . . . . . . 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 476	. . . . . . . . . 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 1397	. . . . . . . . 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 522	. . . . . . . 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 943	. . . . . . 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 1551	. . . . . . 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 175	. . . . . 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 1811	. . . . . 6 |- {(card` A)} = {x | x = (card` A)}
22	20, 21	syl5ss 1544	. . . . 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 1983	. . . . 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 12	. . . 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 2838	. . . . 5 |- (card` A) e. V
26	25	intsn 1991	. . . 4 |- |^|{(card` A)} = (card` A)
27	24, 26	syl6ss 1546	. . 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 1534	. 2 |- (A e. On -> (cf` A) (_ (card` A))
29		0ss 1725	. . 3 |- (/) (_ (card` A)
30		cffnon 3702	. . . . . . . 8 |- cf Fn On
31		fndm 2723	. . . . . . . 8 |- (cf Fn On -> dom cf = On)
32	30, 31	ax-mp 6	. . . . . . 7 |- dom cf = On
33	32	eleq2i 1153	. . . . . 6 |- (A e. dom cf <-> A e. On)
34	33	negbii 162	. . . . 5 |- (-. A e. dom cf <-> -. A e. On)
35		ndmfv 2848	. . . . 5 |- (-. A e. dom cf -> (cf` A) = (/))
36	34, 35	sylbir 176	. . . 4 |- (-. A e. On -> (cf` A) = (/))
37	36	sseq1d 1527	. . 3 |- (-. A e. On -> ((cf` A) (_ (card` A) <-> (/) (_ (card` A)))
38	29, 37	mpbiri 169	. 2 |- (-. A e. On -> (cf` A) (_ (card` A))
39	28, 38	pm2.61i 110	1 |- (cf` A) (_ (card` A)

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196  A.wal 672  E.wex 678  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202   (_ wss 1487  (/)c0 1707  {csn 1808  |^|cint 1965  Oncon0 2199  dom cdm 2410   Fn wfn 2417  ` cfv 2422  cardccrd 3620  cfccf 3622

This theorem is referenced by:  cfle 3708  cfom 3710

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-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-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-int 1966  df-br 2063  df-opab 2098  df-id 2125  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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