Theorem cardval 3633

Description: The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 3661 for a simpler version of its value.

Assertion

Ref	Expression
cardval	|- (card` A) = |^|{x e. On | x ~~ A}

Distinct variable group(s):   x,A

Proof of Theorem cardval

Step	Hyp	Ref	Expression
1		numth2 3600	. . . . 5 |- E.x e. On x ~~ A
2		intexrab 1988	. . . . 5 |- (E.x e. On x ~~ A <-> |^|{x e. On | x ~~ A} e. V)
3	1, 2	mpbi 164	. . . 4 |- |^|{x e. On | x ~~ A} e. V
4		breq2 2066	. . . . . . 7 |- (y = A -> (x ~~ y <-> x ~~ A))
5	4	birabsdv 1344	. . . . . 6 |- (y = A -> {x e. On | x ~~ y} = {x e. On | x ~~ A})
6	5	inteqd 1970	. . . . 5 |- (y = A -> |^|{x e. On | x ~~ y} = |^|{x e. On | x ~~ A})
7	6	fvopabg 2872	. . . 4 |- ((A e. V /\ |^|{x e. On | x ~~ A} e. V) -> ({<.y, z>. | z = |^|{x e. On | x ~~ y}}` A) = |^|{x e. On | x ~~ A})
8	3, 7	mpan2 519	. . 3 |- (A e. V -> ({<.y, z>. | z = |^|{x e. On | x ~~ y}}` A) = |^|{x e. On | x ~~ A})
9		df-card 3623	. . . 4 |- card = {<.y, z>. | z = |^|{x e. On | x ~~ y}}
10	9	fveq1i 2833	. . 3 |- (card` A) = ({<.y, z>. | z = |^|{x e. On | x ~~ y}}` A)
11	8, 10	syl5eq 1136	. 2 |- (A e. V -> (card` A) = |^|{x e. On | x ~~ A})
12		fvprc 2829	. . 3 |- (-. A e. V -> (card` A) = (/))
13		visset 1350	. . . . . . . . . . 11 |- x e. V
14	13	enref 3295	. . . . . . . . . 10 |- x ~~ x
15		brprc 2097	. . . . . . . . . 10 |- (-. A e. V -> (x ~~ A <-> x ~~ x))
16	14, 15	mpbiri 169	. . . . . . . . 9 |- (-. A e. V -> x ~~ A)
17	16	biantrud 545	. . . . . . . 8 |- (-. A e. V -> (x e. On <-> (x e. On /\ x ~~ A)))
18	17	biabdv 1183	. . . . . . 7 |- (-. A e. V -> {x | x e. On} = {x | (x e. On /\ x ~~ A)})
19		df-rab 1208	. . . . . . 7 |- {x e. On | x ~~ A} = {x | (x e. On /\ x ~~ A)}
20	18, 19	syl6reqr 1143	. . . . . 6 |- (-. A e. V -> {x e. On | x ~~ A} = {x | x e. On})
21		abid2 1186	. . . . . 6 |- {x | x e. On} = On
22	20, 21	syl6eq 1140	. . . . 5 |- (-. A e. V -> {x e. On | x ~~ A} = On)
23	22	inteqd 1970	. . . 4 |- (-. A e. V -> |^|{x e. On | x ~~ A} = |^|On)
24		inton 2281	. . . 4 |- |^|On = (/)
25	23, 24	syl6eq 1140	. . 3 |- (-. A e. V -> |^|{x e. On | x ~~ A} = (/))
26	12, 25	eqtr4d 1131	. 2 |- (-. A e. V -> (card` A) = |^|{x e. On | x ~~ A})
27	11, 26	pm2.61i 110	1 |- (card` A) = |^|{x e. On | x ~~ A}

Syntax hints:  -. wn 1   /\ wa 196  {cab 1090   = wceq 1091   e. wcel 1092  E.wrex 1202  {crab 1204  Vcvv 1348  (/)c0 1707  |^|cint 1965   class class class wbr 2054  {copab 2055  Oncon0 2199  ` cfv 2422   ~~ cen 3271  cardccrd 3620

This theorem is referenced by:  cardon 3634  cardid 3635  oncard 3636  cardne 3637  iscard2 3660

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  ax-reg 1078  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  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-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  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-tp 1814  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-suc 2205  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-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-en 3274  df-card 3623

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