Theorem cardcf 4894

Description: Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103.

Assertion

Ref	Expression
cardcf	|- (card` (cf` A)) = (cf` A)

Proof of Theorem cardcf

Step	Hyp	Ref	Expression
1		cfval 4889	. . . 4 |- (A e. On -> (cf` A) = |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
2		fvex 3727	. . . . . . 7 |- (cf` A) e. V
3	1, 2	syl6eqelr 1555	. . . . . 6 |- (A e. On -> |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} e. V)
4		intex 2725	. . . . . 6 |- ({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))} e. V)
5	3, 4	sylibr 200	. . . . 5 |- (A e. On -> {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} =/= (/))
6		visset 1810	. . . . . . . . . 10 |- v e. V
7		eqeq1 1479	. . . . . . . . . . . 12 |- (x = v -> (x = (card` y) <-> v = (card` y)))
8	7	anbi1d 616	. . . . . . . . . . 11 |- (x = v -> ((x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) <-> (v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
9	8	exbidv 1278	. . . . . . . . . 10 |- (x = v -> (E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) <-> E.y(v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
10	6, 9	elab 1894	. . . . . . . . 9 |- (v e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} <-> E.y(v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)))
11		fveq2 3719	. . . . . . . . . . . . 13 |- (v = (card` y) -> (card` v) = (card` (card` y)))
12		cardidm 4832	. . . . . . . . . . . . 13 |- (card` (card` y)) = (card` y)
13	11, 12	syl6eq 1521	. . . . . . . . . . . 12 |- (v = (card` y) -> (card` v) = (card` y))
14		eqeq2 1482	. . . . . . . . . . . 12 |- (v = (card` y) -> ((card` v) = v <-> (card` v) = (card` y)))
15	13, 14	mpbird 196	. . . . . . . . . . 11 |- (v = (card` y) -> (card` v) = v)
16	15	adantr 389	. . . . . . . . . 10 |- ((v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) -> (card` v) = v)
17	16	19.23aiv 1294	. . . . . . . . 9 |- (E.y(v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) -> (card` v) = v)
18	10, 17	sylbi 199	. . . . . . . 8 |- (v e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} -> (card` v) = v)
19		cardon 4810	. . . . . . . 8 |- (card` v) e. On
20	18, 19	syl6eqelr 1555	. . . . . . 7 |- (v e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} -> v e. On)
21	20	ssriv 2066	. . . . . 6 |- {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} (_ On
22		onint 3002	. . . . . 6 |- (({x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} (_ 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.z e. A E.w e. y z (_ w))} e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
23	21, 22	mpan 694	. . . . 5 |- ({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))} e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
24	5, 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))} e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
25	1, 24	eqeltrd 1546	. . 3 |- (A e. On -> (cf` A) e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
26		fveq2 3719	. . . . 5 |- (v = (cf` A) -> (card` v) = (card` (cf` A)))
27		id 59	. . . . 5 |- (v = (cf` A) -> v = (cf` A))
28	26, 27	eqeq12d 1487	. . . 4 |- (v = (cf` A) -> ((card` v) = v <-> (card` (cf` A)) = (cf` A)))
29	28, 18	vtoclga 1849	. . 3 |- ((cf` A) e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} -> (card` (cf` A)) = (cf` A))
30	25, 29	syl 10	. 2 |- (A e. On -> (card` (cf` A)) = (cf` A))
31		cffnon 4890	. . . . . . 7 |- cf Fn On
32		fndm 3583	. . . . . . 7 |- (cf Fn On -> dom cf = On)
33	31, 32	ax-mp 7	. . . . . 6 |- dom cf = On
34	33	eleq2i 1536	. . . . 5 |- (A e. dom cf <-> A e. On)
35	34	negbii 187	. . . 4 |- (-. A e. dom cf <-> -. A e. On)
36		ndmfv 3740	. . . 4 |- (-. A e. dom cf -> (cf` A) = (/))
37	35, 36	sylbir 201	. . 3 |- (-. A e. On -> (cf` A) = (/))
38		card0 4806	. . . 4 |- (card` (/)) = (/)
39		fveq2 3719	. . . 4 |- ((cf` A) = (/) -> (card` (cf` A)) = (card` (/)))
40		id 59	. . . 4 |- ((cf` A) = (/) -> (cf` A) = (/))
41	38, 39, 40	3eqtr4a 1530	. . 3 |- ((cf` A) = (/) -> (card` (cf` A)) = (cf` A))
42	37, 41	syl 10	. 2 |- (-. A e. On -> (card` (cf` A)) = (cf` A))
43	30, 42	pm2.61i 126	1 |- (card` (cf` A)) = (cf` A)

Syntax hints:  -. wn 2   /\ wa 223   = wceq 955   e. wcel 957  E.wex 979  {cab 1462   =/= wne 1583  A.wral 1643  E.wrex 1644  Vcvv 1808   (_ wss 2044  (/)c0 2277  |^|cint 2529  Oncon0 2944  dom cdm 3166   Fn wfn 3173  ` cfv 3178  cardccrd 4796  cfccf 4798

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-ac 4727

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-suc 2950  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-er 4254  df-en 4360  df-card 4799  df-cf 4801

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