Theorem rankval 4668

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 4644	. . 3 |- rank = {<.y, z>. | z = |^|{x e. On | y e. (R1` suc x)}}
2	1	fveq1i 3725	. 2 |- (rank` A) = ({<.y, z>. | z = |^|{x e. On | y e. (R1` suc x)}}` A)
3		rankval.1	. . 3 |- A e. V
4		rankwflem 4665	. . . . . 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 2292	. . . . 5 |- ({x e. On | A e. (R1` suc x)} =/= (/) <-> E.x e. On A e. (R1` suc x))
7	5, 6	mpbir 190	. . . 4 |- {x e. On | A e. (R1` suc x)} =/= (/)
8		intex 2729	. . . 4 |- ({x e. On | A e. (R1` suc x)} =/= (/) <-> |^|{x e. On | A e. (R1` suc x)} e. V)
9	7, 8	mpbi 189	. . 3 |- |^|{x e. On | A e. (R1` suc x)} e. V
10		eleq1 1534	. . . . 5 |- (y = A -> (y e. (R1` suc x) <-> A e. (R1` suc x)))
11	10	rabbisdv 1807	. . . 4 |- (y = A -> {x e. On | y e. (R1` suc x)} = {x e. On | A e. (R1` suc x)})
12	11	inteqd 2538	. . 3 |- (y = A -> |^|{x e. On | y e. (R1` suc x)} = |^|{x e. On | A e. (R1` suc x)})
13	3, 9, 12	fvopab 3790	. 2 |- ({<.y, z>. | z = |^|{x e. On | y e. (R1` suc x)}}` A) = |^|{x e. On | A e. (R1` suc x)}
14	2, 13	eqtr 1495	1 |- (rank` A) = |^|{x e. On | A e. (R1` suc x)}

Syntax hints:   = wceq 956   e. wcel 958   =/= wne 1585  E.wrex 1646  {crab 1648  Vcvv 1811  (/)c0 2280  |^|cint 2533  {copab 2666  Oncon0 2948  suc csuc 2950  ` cfv 3182  R1cr1 4641  rankcrnk 4642

This theorem is referenced by:  rankvalg 4669  rankid 4672  rankr1 4674  rankval3 4681

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-reg 4593  ax-inf2 4625

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-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  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-lim 2953  df-suc 2954  df-om 3132  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-fv 3198  df-rdg 3932  df-r1 4643  df-rank 4644

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