Theorem cardval 4806

Description: The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 4835 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 4765	. . . . 5 |- E.x e. On x ~~ A
2		intexrab 2727	. . . . 5 |- (E.x e. On x ~~ A <-> |^|{x e. On | x ~~ A} e. V)
3	1, 2	mpbi 189	. . . 4 |- |^|{x e. On | x ~~ A} e. V
4		breq2 2618	. . . . . . 7 |- (y = A -> (x ~~ y <-> x ~~ A))
5	4	rabbisdv 1803	. . . . . 6 |- (y = A -> {x e. On | x ~~ y} = {x e. On | x ~~ A})
6	5	inteqd 2533	. . . . 5 |- (y = A -> |^|{x e. On | x ~~ y} = |^|{x e. On | x ~~ A})
7	6	fvopabg 3776	. . . 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 695	. . 3 |- (A e. V -> ({<.y, z>. | z = |^|{x e. On | x ~~ y}}` A) = |^|{x e. On | x ~~ A})
9		df-card 4796	. . . 4 |- card = {<.y, z>. | z = |^|{x e. On | x ~~ y}}
10	9	fveq1i 3716	. . 3 |- (card` A) = ({<.y, z>. | z = |^|{x e. On | x ~~ y}}` A)
11	8, 10	syl5eq 1516	. 2 |- (A e. V -> (card` A) = |^|{x e. On | x ~~ A})
12		fvprc 3712	. . 3 |- (-. A e. V -> (card` A) = (/))
13		visset 1809	. . . . . . . . . . 11 |- x e. V
14	13	enref 4378	. . . . . . . . . 10 |- x ~~ x
15		brprc 2656	. . . . . . . . . 10 |- (-. A e. V -> (x ~~ A <-> x ~~ x))
16	14, 15	mpbiri 194	. . . . . . . . 9 |- (-. A e. V -> x ~~ A)
17	16	biantrud 725	. . . . . . . 8 |- (-. A e. V -> (x e. On <-> (x e. On /\ x ~~ A)))
18	17	abbidv 1574	. . . . . . 7 |- (-. A e. V -> {x | x e. On} = {x | (x e. On /\ x ~~ A)})
19		df-rab 1649	. . . . . . 7 |- {x e. On | x ~~ A} = {x | (x e. On /\ x ~~ A)}
20	18, 19	syl6reqr 1523	. . . . . 6 |- (-. A e. V -> {x e. On | x ~~ A} = {x | x e. On})
21		abid2 1577	. . . . . 6 |- {x | x e. On} = On
22	20, 21	syl6eq 1520	. . . . 5 |- (-. A e. V -> {x e. On | x ~~ A} = On)
23	22	inteqd 2533	. . . 4 |- (-. A e. V -> |^|{x e. On | x ~~ A} = |^|On)
24		inton 3021	. . . 4 |- |^|On = (/)
25	23, 24	syl6eq 1520	. . 3 |- (-. A e. V -> |^|{x e. On | x ~~ A} = (/))
26	12, 25	eqtr4d 1507	. 2 |- (-. A e. V -> (card` A) = |^|{x e. On | x ~~ A})
27	11, 26	pm2.61i 126	1 |- (card` A) = |^|{x e. On | x ~~ A}

Syntax hints:  -. wn 2   /\ wa 223   = wceq 954   e. wcel 956  {cab 1461  E.wrex 1643  {crab 1645  Vcvv 1807  (/)c0 2276  |^|cint 2528   class class class wbr 2614  {copab 2661  Oncon0 2943  ` cfv 3177   ~~ cen 4354  cardccrd 4793

This theorem is referenced by:  cardon 4807  cardid 4808  oncard 4809  cardne 4810  iscard2 4834

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-ac 4724

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-suc 2949  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-en 4357  df-card 4796

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