Theorem cardval 5975

Description: The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 6007 for a simpler version of its value.

Assertion

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

Distinct variable group:   x,A

Proof of Theorem cardval

Step	Hyp	Ref	Expression
1		numth2 5947	. . . . 5 |- E.x e. On x ~~ A
2		intexrab 3468	. . . . 5 |- (E.x e. On x ~~ A <-> |^|{x e. On | x ~~ A} e. _V)
3	1, 2	mpbi 206	. . . 4 |- |^|{x e. On | x ~~ A} e. _V
4		breq2 3342	. . . . . . 7 |- (y = A -> (x ~~ y <-> x ~~ A))
5	4	rabbidv 2287	. . . . . 6 |- (y = A -> {x e. On | x ~~ y} = {x e. On | x ~~ A})
6	5	inteqd 3219	. . . . 5 |- (y = A -> |^|{x e. On | x ~~ y} = |^|{x e. On | x ~~ A})
7	6	fvopabg 4748	. . . 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 760	. . 3 |- (A e. _V -> ({<.y, z>. | z = |^|{x e. On | x ~~ y}}` A) = |^|{x e. On | x ~~ A})
9		df-card 5862	. . . 4 |- card = {<.y, z>. | z = |^|{x e. On | x ~~ y}}
10	9	fveq1i 4682	. . 3 |- (card` A) = ({<.y, z>. | z = |^|{x e. On | x ~~ y}}` A)
11	8, 10	syl5eq 1940	. 2 |- (A e. _V -> (card` A) = |^|{x e. On | x ~~ A})
12		fvprc 4678	. . 3 |- (-. A e. _V -> (card` A) = (/))
13		visset 2295	. . . . . . . . . . 11 |- x e. _V
14	13	enref 5450	. . . . . . . . . 10 |- x ~~ x
15		brprc 3386	. . . . . . . . . 10 |- (-. A e. _V -> (x ~~ A <-> x ~~ x))
16	14, 15	mpbiri 211	. . . . . . . . 9 |- (-. A e. _V -> x ~~ A)
17	16	biantrud 795	. . . . . . . 8 |- (-. A e. _V -> (x e. On <-> (x e. On /\ x ~~ A)))
18	17	abbidv 2008	. . . . . . 7 |- (-. A e. _V -> {x | x e. On} = {x | (x e. On /\ x ~~ A)})
19		df-rab 2112	. . . . . . 7 |- {x e. On | x ~~ A} = {x | (x e. On /\ x ~~ A)}
20	18, 19	syl6reqr 1947	. . . . . 6 |- (-. A e. _V -> {x e. On | x ~~ A} = {x | x e. On})
21		abid2 2011	. . . . . 6 |- {x | x e. On} = On
22	20, 21	syl6eq 1944	. . . . 5 |- (-. A e. _V -> {x e. On | x ~~ A} = On)
23	22	inteqd 3219	. . . 4 |- (-. A e. _V -> |^|{x e. On | x ~~ A} = |^|On)
24		inton 3720	. . . 4 |- |^|On = (/)
25	23, 24	syl6eq 1944	. . 3 |- (-. A e. _V -> |^|{x e. On | x ~~ A} = (/))
26	12, 25	eqtr4d 1928	. 2 |- (-. A e. _V -> (card` A) = |^|{x e. On | x ~~ A})
27	11, 26	pm2.61i 140	1 |- (card` A) = |^|{x e. On | x ~~ A}

Syntax hints:  -. wn 2   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  E.wrex 2106  {crab 2108  _Vcvv 2292  (/)c0 2875  |^|cint 3214   class class class wbr 3338  {copab 3395  Oncon0 3657  ` cfv 3998   ~~ cen 5423  cardccrd 5859

This theorem is referenced by:  cardon 5976  cardid 5977  oncard 5978  cardne 5980  iscard2 6006

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-en 5427  df-card 5862

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