Theorem cfub 3703

Description: An upper bound on cofinality.

Assertion

Ref	Expression
cfub	|- (cf` A) (_ |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y))}

Distinct variable group(s):   x,y,A

Proof of Theorem cfub

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		ssel 1502	. . . . . . . . . . . . . . . . . 18 |- (y (_ A -> (w e. y -> w e. A))
3		onelon 2223	. . . . . . . . . . . . . . . . . . 19 |- ((A e. On /\ w e. A) -> w e. On)
4	3	exp 291	. . . . . . . . . . . . . . . . . 18 |- (A e. On -> (w e. A -> w e. On))
5	2, 4	sylan9r 360	. . . . . . . . . . . . . . . . 17 |- ((A e. On /\ y (_ A) -> (w e. y -> w e. On))
6		onelsst 2255	. . . . . . . . . . . . . . . . 17 |- (w e. On -> (z e. w -> z (_ w))
7	5, 6	syl6 23	. . . . . . . . . . . . . . . 16 |- ((A e. On /\ y (_ A) -> (w e. y -> (z e. w -> z (_ w)))
8	7	imdistand 342	. . . . . . . . . . . . . . 15 |- ((A e. On /\ y (_ A) -> ((w e. y /\ z e. w) -> (w e. y /\ z (_ w)))
9	8	ancomsd 335	. . . . . . . . . . . . . 14 |- ((A e. On /\ y (_ A) -> ((z e. w /\ w e. y) -> (w e. y /\ z (_ w)))
10	9	19.22dv 947	. . . . . . . . . . . . 13 |- ((A e. On /\ y (_ A) -> (E.w(z e. w /\ w e. y) -> E.w(w e. y /\ z (_ w)))
11		eluni 1922	. . . . . . . . . . . . 13 |- (z e. U.y <-> E.w(z e. w /\ w e. y))
12		df-rex 1206	. . . . . . . . . . . . 13 |- (E.w e. y z (_ w <-> E.w(w e. y /\ z (_ w))
13	10, 11, 12	3imtr4g 426	. . . . . . . . . . . 12 |- ((A e. On /\ y (_ A) -> (z e. U.y -> E.w e. y z (_ w))
14	13	r19.20sdv 1257	. . . . . . . . . . 11 |- ((A e. On /\ y (_ A) -> (A.z e. A z e. U.y -> A.z e. A E.w e. y z (_ w))
15		dfss3 1498	. . . . . . . . . . 11 |- (A (_ U.y <-> A.z e. A z e. U.y)
16	14, 15	syl5ib 181	. . . . . . . . . 10 |- ((A e. On /\ y (_ A) -> (A (_ U.y -> A.z e. A E.w e. y z (_ w))
17	16	exp 291	. . . . . . . . 9 |- (A e. On -> (y (_ A -> (A (_ U.y -> A.z e. A E.w e. y z (_ w)))
18	17	imdistand 342	. . . . . . . 8 |- (A e. On -> ((y (_ A /\ A (_ U.y) -> (y (_ A /\ A.z e. A E.w e. y z (_ w)))
19	18	anim2d 433	. . . . . . 7 |- (A e. On -> ((x = (card` y) /\ (y (_ A /\ A (_ U.y)) -> (x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
20	19	19.22dv 947	. . . . . 6 |- (A e. On -> (E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y)) -> E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
21	20	19.21aiv 943	. . . . 5 |- (A e. On -> A.x(E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y)) -> E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
22		ss2ab 1551	. . . . 5 |- ({x | E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y))} (_ {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} <-> A.x(E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y)) -> E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
23	21, 22	sylibr 175	. . . 4 |- (A e. On -> {x | E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y))} (_ {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
24		intss 1983	. . . 4 |- ({x | E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y))} (_ {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))} (_ |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y))})
25	23, 24	syl 12	. . 3 |- (A e. 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 (_ U.y))})
26	1, 25	eqsstrd 1534	. 2 |- (A e. On -> (cf` A) (_ |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y))})
27		0ss 1725	. . 3 |- (/) (_ |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y))}
28		cffnon 3702	. . . . . . . 8 |- cf Fn On
29		fndm 2723	. . . . . . . 8 |- (cf Fn On -> dom cf = On)
30	28, 29	ax-mp 6	. . . . . . 7 |- dom cf = On
31	30	eleq2i 1153	. . . . . 6 |- (A e. dom cf <-> A e. On)
32	31	negbii 162	. . . . 5 |- (-. A e. dom cf <-> -. A e. On)
33		ndmfv 2848	. . . . 5 |- (-. A e. dom cf -> (cf` A) = (/))
34	32, 33	sylbir 176	. . . 4 |- (-. A e. On -> (cf` A) = (/))
35	34	sseq1d 1527	. . 3 |- (-. A e. On -> ((cf` A) (_ |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y))} <-> (/) (_ |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y))}))
36	27, 35	mpbiri 169	. 2 |- (-. A e. On -> (cf` A) (_ |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y))})
37	26, 36	pm2.61i 110	1 |- (cf` A) (_ |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y))}

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   e. wel 803  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202   (_ wss 1487  (/)c0 1707  U.cuni 1919  |^|cint 1965  Oncon0 2199  dom cdm 2410   Fn wfn 2417  ` cfv 2422  cardccrd 3620  cfccf 3622

This theorem is referenced by:  cflim 3704  cf0 3705

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-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-pw 1799  df-sn 1811  df-pr 1812  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-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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