Theorem rankid 4644

Description: Identity law for the rank function.

Hypothesis

Ref	Expression
rankid.1	|- A e. V

Assertion

Ref	Expression
rankid	|- A e. (R1` suc (rank` A))

Proof of Theorem rankid

Step	Hyp	Ref	Expression
1		rankid.1	. . . 4 |- A e. V
2		rankwflem 4637	. . . 4 |- (A e. V -> E.x e. On A e. (R1` suc x))
3	1, 2	ax-mp 7	. . 3 |- E.x e. On A e. (R1` suc x)
4		ax-17 968	. . . . 5 |- (y e. A -> A.x y e. A)
5		ax-17 968	. . . . . 6 |- (y e. R1 -> A.x y e. R1)
6		hbrab1 1764	. . . . . . . 8 |- (y e. {x e. On | A e. (R1` suc x)} -> A.x y e. {x e. On | A e. (R1` suc x)})
7	6	hbint 2533	. . . . . . 7 |- (y e. |^|{x e. On | A e. (R1` suc x)} -> A.x y e. |^|{x e. On | A e. (R1` suc x)})
8	7	hbsuc 3030	. . . . . 6 |- (y e. suc |^|{x e. On | A e. (R1` suc x)} -> A.x y e. suc |^|{x e. On | A e. (R1` suc x)})
9	5, 8	hbfv 3714	. . . . 5 |- (y e. (R1` suc |^|{x e. On | A e. (R1` suc x)}) -> A.x y e. (R1` suc |^|{x e. On | A e. (R1` suc x)}))
10	4, 9	hbel 1558	. . . 4 |- (A e. (R1` suc |^|{x e. On | A e. (R1` suc x)}) -> A.x A e. (R1` suc |^|{x e. On | A e. (R1` suc x)}))
11		suceq 3024	. . . . . 6 |- (x = |^|{x e. On | A e. (R1` suc x)} -> suc x = suc |^|{x e. On | A e. (R1` suc x)})
12	11	fveq2d 3713	. . . . 5 |- (x = |^|{x e. On | A e. (R1` suc x)} -> (R1` suc x) = (R1` suc |^|{x e. On | A e. (R1` suc x)}))
13	12	eleq2d 1533	. . . 4 |- (x = |^|{x e. On | A e. (R1` suc x)} -> (A e. (R1` suc x) <-> A e. (R1` suc |^|{x e. On | A e. (R1` suc x)})))
14	10, 13	onminsb 2999	. . 3 |- (E.x e. On A e. (R1` suc x) -> A e. (R1` suc |^|{x e. On | A e. (R1` suc x)}))
15	3, 14	ax-mp 7	. 2 |- A e. (R1` suc |^|{x e. On | A e. (R1` suc x)})
16	1	rankval 4640	. . . 4 |- (rank` A) = |^|{x e. On | A e. (R1` suc x)}
17		suceq 3024	. . . 4 |- ((rank` A) = |^|{x e. On | A e. (R1` suc x)} -> suc (rank` A) = suc |^|{x e. On | A e. (R1` suc x)})
18	16, 17	ax-mp 7	. . 3 |- suc (rank` A) = suc |^|{x e. On | A e. (R1` suc x)}
19	18	fveq2i 3712	. 2 |- (R1` suc (rank` A)) = (R1` suc |^|{x e. On | A e. (R1` suc x)})
20	15, 19	eleqtrr 1539	1 |- A e. (R1` suc (rank` A))

Syntax hints:   = wceq 953   e. wcel 955  E.wrex 1638  {crab 1640  Vcvv 1802  |^|cint 2523  Oncon0 2938  suc csuc 2940  ` cfv 3172  R1cr1 4613  rankcrnk 4614

This theorem is referenced by:  rankr1lem 4645  rankel 4652  rankval3 4653  bndrank 4654  rankpw 4656  rankval4 4674

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917  df-r1 4615  df-rank 4616

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