Theorem cardval 4826

Description: The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 4855 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 4785	. . . . 5 |- E.x e. On x ~~ A
2		intexrab 2732	. . . . 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 2623	. . . . . . 7 |- (y = A -> (x ~~ y <-> x ~~ A))
5	4	rabbisdv 1807	. . . . . 6 |- (y = A -> {x e. On | x ~~ y} = {x e. On | x ~~ A})
6	5	inteqd 2538	. . . . 5 |- (y = A -> |^|{x e. On | x ~~ y} = |^|{x e. On | x ~~ A})
7	6	fvopabg 3785	. . . 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 696	. . 3 |- (A e. V -> ({<.y, z>. | z = |^|{x e. On | x ~~ y}}` A) = |^|{x e. On | x ~~ A})
9		df-card 4816	. . . 4 |- card = {<.y, z>. | z = |^|{x e. On | x ~~ y}}
10	9	fveq1i 3725	. . 3 |- (card` A) = ({<.y, z>. | z = |^|{x e. On | x ~~ y}}` A)
11	8, 10	syl5eq 1519	. 2 |- (A e. V -> (card` A) = |^|{x e. On | x ~~ A})
12		fvprc 3721	. . 3 |- (-. A e. V -> (card` A) = (/))
13		visset 1813	. . . . . . . . . . 11 |- x e. V
14	13	enref 4391	. . . . . . . . . 10 |- x ~~ x
15		brprc 2661	. . . . . . . . . 10 |- (-. A e. V -> (x ~~ A <-> x ~~ x))
16	14, 15	mpbiri 194	. . . . . . . . 9 |- (-. A e. V -> x ~~ A)
17	16	biantrud 726	. . . . . . . 8 |- (-. A e. V -> (x e. On <-> (x e. On /\ x ~~ A)))
18	17	abbidv 1577	. . . . . . 7 |- (-. A e. V -> {x | x e. On} = {x | (x e. On /\ x ~~ A)})
19		df-rab 1652	. . . . . . 7 |- {x e. On | x ~~ A} = {x | (x e. On /\ x ~~ A)}
20	18, 19	syl6reqr 1526	. . . . . 6 |- (-. A e. V -> {x e. On | x ~~ A} = {x | x e. On})
21		abid2 1580	. . . . . 6 |- {x | x e. On} = On
22	20, 21	syl6eq 1523	. . . . 5 |- (-. A e. V -> {x e. On | x ~~ A} = On)
23	22	inteqd 2538	. . . 4 |- (-. A e. V -> |^|{x e. On | x ~~ A} = |^|On)
24		inton 3026	. . . 4 |- |^|On = (/)
25	23, 24	syl6eq 1523	. . 3 |- (-. A e. V -> |^|{x e. On | x ~~ A} = (/))
26	12, 25	eqtr4d 1510	. 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 956   e. wcel 958  {cab 1463  E.wrex 1646  {crab 1648  Vcvv 1811  (/)c0 2280  |^|cint 2533   class class class wbr 2619  {copab 2666  Oncon0 2948  ` cfv 3182   ~~ cen 4364  cardccrd 4813

This theorem is referenced by:  cardon 4827  cardid 4828  oncard 4829  cardne 4830  iscard2 4854

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-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  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-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  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-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  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-suc 2954  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-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-en 4368  df-card 4816

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