Theorem rankval 5779

Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition).

Hypothesis

Ref	Expression
rankval.1	|- A e. _V

Assertion

Ref	Expression
rankval	|- (rank` A) = |^|{x e. On | A e. (R1` suc x)}

Distinct variable group:   x,A

Proof of Theorem rankval

Step	Hyp	Ref	Expression
1		df-rank 5751	. . 3 |- rank = {<.y, z>. | z = |^|{x e. On | y e. (R1` suc x)}}
2	1	fveq1i 4682	. 2 |- (rank` A) = ({<.y, z>. | z = |^|{x e. On | y e. (R1` suc x)}}` A)
3		rankval.1	. . 3 |- A e. _V
4		rankwflem 5776	. . . . . 6 |- (A e. _V -> E.x e. On A e. (R1` suc x))
5	3, 4	ax-mp 7	. . . . 5 |- E.x e. On A e. (R1` suc x)
6		rabn0 2893	. . . . 5 |- ({x e. On | A e. (R1` suc x)} =/= (/) <-> E.x e. On A e. (R1` suc x))
7	5, 6	mpbir 207	. . . 4 |- {x e. On | A e. (R1` suc x)} =/= (/)
8		intex 3465	. . . 4 |- ({x e. On | A e. (R1` suc x)} =/= (/) <-> |^|{x e. On | A e. (R1` suc x)} e. _V)
9	7, 8	mpbi 206	. . 3 |- |^|{x e. On | A e. (R1` suc x)} e. _V
10		eleq1 1957	. . . . 5 |- (y = A -> (y e. (R1` suc x) <-> A e. (R1` suc x)))
11	10	rabbidv 2287	. . . 4 |- (y = A -> {x e. On | y e. (R1` suc x)} = {x e. On | A e. (R1` suc x)})
12	11	inteqd 3219	. . 3 |- (y = A -> |^|{x e. On | y e. (R1` suc x)} = |^|{x e. On | A e. (R1` suc x)})
13	3, 9, 12	fvopab 4753	. 2 |- ({<.y, z>. | z = |^|{x e. On | y e. (R1` suc x)}}` A) = |^|{x e. On | A e. (R1` suc x)}
14	2, 13	eqtri 1908	1 |- (rank` A) = |^|{x e. On | A e. (R1` suc x)}

Syntax hints:   = wceq 1298   e. wcel 1300   =/= wne 2017  E.wrex 2106  {crab 2108  _Vcvv 2292  (/)c0 2875  |^|cint 3214  {copab 3395  Oncon0 3657  suc csuc 3659  ` cfv 3998  R1cr1 5748  rankcrnk 5749

This theorem is referenced by:  rankvalg 5780  rankid 5783  rankr1 5785  rankval3 5792

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-reg 5695  ax-inf2 5731

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-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-if 2983  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-lim 3662  df-suc 3663  df-om 3950  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-fv 4014  df-rdg 5140  df-r1 5750  df-rank 5751

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