Theorem rankval 3512

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 ∈ V

Assertion

Ref	Expression
rankval	⊢ (rank ‘A) = ∩{x ∈ On∣A ∈ (R1 ‘suc x)}

Distinct variable group(s):   x,A

Proof of Theorem rankval

Step	Hyp	Ref	Expression
1		df-rank 3488	. . 3 ⊢ rank = {⟨y, z⟩∣z = ∩{x ∈ On∣y ∈ (R1 ‘suc x)}}
2	1	fveq1i 2833	. 2 ⊢ (rank ‘A) = ({⟨y, z⟩∣z = ∩{x ∈ On∣y ∈ (R1 ‘suc x)}} ‘A)
3		rankval.1	. . 3 ⊢ A ∈ V
4		rankwflem 3509	. . . . . 6 ⊢ (A ∈ V → ∃x ∈ On A ∈ (R1 ‘suc x))
5	3, 4	ax-mp 6	. . . . 5 ⊢ ∃x ∈ On A ∈ (R1 ‘suc x)
6		rabn0 1716	. . . . 5 ⊢ (¬ {x ∈ On∣A ∈ (R1 ‘suc x)} = ∅ ↔ ∃x ∈ On A ∈ (R1 ‘suc x))
7	5, 6	mpbir 165	. . . 4 ⊢ ¬ {x ∈ On∣A ∈ (R1 ‘suc x)} = ∅
8		intex 1986	. . . 4 ⊢ (¬ {x ∈ On∣A ∈ (R1 ‘suc x)} = ∅ ↔ ∩{x ∈ On∣A ∈ (R1 ‘suc x)} ∈ V)
9	7, 8	mpbi 164	. . 3 ⊢ ∩{x ∈ On∣A ∈ (R1 ‘suc x)} ∈ V
10		eleq1 1149	. . . . 5 ⊢ (y = A → (y ∈ (R1 ‘suc x) ↔ A ∈ (R1 ‘suc x)))
11	10	birabsdv 1344	. . . 4 ⊢ (y = A → {x ∈ On∣y ∈ (R1 ‘suc x)} = {x ∈ On∣A ∈ (R1 ‘suc x)})
12	11	inteqd 1970	. . 3 ⊢ (y = A → ∩{x ∈ On∣y ∈ (R1 ‘suc x)} = ∩{x ∈ On∣A ∈ (R1 ‘suc x)})
13	3, 9, 12	fvopab 2877	. 2 ⊢ ({⟨y, z⟩∣z = ∩{x ∈ On∣y ∈ (R1 ‘suc x)}} ‘A) = ∩{x ∈ On∣A ∈ (R1 ‘suc x)}
14	2, 13	eqtr 1119	1 ⊢ (rank ‘A) = ∩{x ∈ On∣A ∈ (R1 ‘suc x)}

Syntax hints:  ¬ wn 1   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  {crab 1204  Vcvv 1348  ∅c0 1707  ∩cint 1965  {copab 2055  Oncon0 2199  suc csuc 2201   ‘cfv 2422  R₁cr1 3485  rankcrnk 3486

This theorem is referenced by:  rankvalg 3513  rankid 3516  rankr1 3518  rankval3 3525

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487  df-rank 3488

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