Theorem cfub 4900

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:   x,y,A

Proof of Theorem cfub

Step	Hyp	Ref	Expression
1		cfval 4898	. . 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 2063	. . . . . . . . . . . . . . . . . 18 |- (y (_ A -> (w e. y -> w e. A))
3		onelon 2972	. . . . . . . . . . . . . . . . . . 19 |- ((A e. On /\ w e. A) -> w e. On)
4	3	ex 373	. . . . . . . . . . . . . . . . . 18 |- (A e. On -> (w e. A -> w e. On))
5	2, 4	sylan9r 469	. . . . . . . . . . . . . . . . 17 |- ((A e. On /\ y (_ A) -> (w e. y -> w e. On))
6		onelsst 3000	. . . . . . . . . . . . . . . . 17 |- (w e. On -> (z e. w -> z (_ w))
7	5, 6	syl6 22	. . . . . . . . . . . . . . . 16 |- ((A e. On /\ y (_ A) -> (w e. y -> (z e. w -> z (_ w)))
8	7	imdistand 445	. . . . . . . . . . . . . . 15 |- ((A e. On /\ y (_ A) -> ((w e. y /\ z e. w) -> (w e. y /\ z (_ w)))
9	8	ancomsd 437	. . . . . . . . . . . . . 14 |- ((A e. On /\ y (_ A) -> ((z e. w /\ w e. y) -> (w e. y /\ z (_ w)))
10	9	19.22dv 1290	. . . . . . . . . . . . 13 |- ((A e. On /\ y (_ A) -> (E.w(z e. w /\ w e. y) -> E.w(w e. y /\ z (_ w)))
11		eluni 2506	. . . . . . . . . . . . 13 |- (z e. U.y <-> E.w(z e. w /\ w e. y))
12		df-rex 1650	. . . . . . . . . . . . 13 |- (E.w e. y z (_ w <-> E.w(w e. y /\ z (_ w))
13	10, 11, 12	3imtr4g 553	. . . . . . . . . . . 12 |- ((A e. On /\ y (_ A) -> (z e. U.y -> E.w e. y z (_ w))
14	13	r19.20sdv 1710	. . . . . . . . . . 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 2059	. . . . . . . . . . 11 |- (A (_ U.y <-> A.z e. A z e. U.y)
16	14, 15	syl5ib 206	. . . . . . . . . 10 |- ((A e. On /\ y (_ A) -> (A (_ U.y -> A.z e. A E.w e. y z (_ w))
17	16	ex 373	. . . . . . . . 9 |- (A e. On -> (y (_ A -> (A (_ U.y -> A.z e. A E.w e. y z (_ w)))
18	17	imdistand 445	. . . . . . . 8 |- (A e. On -> ((y (_ A /\ A (_ U.y) -> (y (_ A /\ A.z e. A E.w e. y z (_ w)))
19	18	anim2d 561	. . . . . . 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 1290	. . . . . 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 1286	. . . . 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 2116	. . . . 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 200	. . . 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 2554	. . . 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 10	. . 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 2095	. 2 |- (A e. On -> (cf` A) (_ |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y))})
27		0ss 2301	. . 3 |- (/) (_ |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y))}
28		cffnon 4899	. . . . . . . 8 |- cf Fn On
29		fndm 3587	. . . . . . . 8 |- (cf Fn On -> dom cf = On)
30	28, 29	ax-mp 7	. . . . . . 7 |- dom cf = On
31	30	eleq2i 1538	. . . . . 6 |- (A e. dom cf <-> A e. On)
32	31	negbii 187	. . . . 5 |- (-. A e. dom cf <-> -. A e. On)
33		ndmfv 3745	. . . . 5 |- (-. A e. dom cf -> (cf` A) = (/))
34	32, 33	sylbir 201	. . . 4 |- (-. A e. On -> (cf` A) = (/))
35	34	sseq1d 2088	. . 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 194	. 2 |- (-. A e. On -> (cf` A) (_ |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y))})
37	26, 36	pm2.61i 126	1 |- (cf` A) (_ |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y))}

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  A.wral 1645  E.wrex 1646   (_ wss 2047  (/)c0 2280  U.cuni 2503  |^|cint 2533  Oncon0 2948  dom cdm 3170   Fn wfn 3177  ` cfv 3182  cardccrd 4805  cfccf 4807

This theorem is referenced by:  cflim 4901  cf0 4902

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 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-int 2534  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-cf 4810

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