Theorem rankid 3516

Description: Identity law for the rank function.

Hypothesis

Ref	Expression
rankid.1	⊢ A ∈ V

Assertion

Ref	Expression
rankid	⊢ A ∈ (R1 ‘suc (rank ‘A))

Proof of Theorem rankid

Step	Hyp	Ref	Expression
1		rankid.1	. . . 4 ⊢ A ∈ V
2		rankwflem 3509	. . . 4 ⊢ (A ∈ V → ∃x ∈ On A ∈ (R1 ‘suc x))
3	1, 2	ax-mp 6	. . 3 ⊢ ∃x ∈ On A ∈ (R1 ‘suc x)
4		ax-17 925	. . . . 5 ⊢ (y ∈ A → ∀x y ∈ A)
5		ax-17 925	. . . . . 6 ⊢ (y ∈ R1 → ∀x y ∈ R1)
6		hbrab1 1310	. . . . . . . 8 ⊢ (y ∈ {x ∈ On∣A ∈ (R1 ‘suc x)} → ∀x y ∈ {x ∈ On∣A ∈ (R1 ‘suc x)})
7	6	hbint 1975	. . . . . . 7 ⊢ (y ∈ ∩{x ∈ On∣A ∈ (R1 ‘suc x)} → ∀x y ∈ ∩{x ∈ On∣A ∈ (R1 ‘suc x)})
8	7	hbsuc 2294	. . . . . 6 ⊢ (y ∈ suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)} → ∀x y ∈ suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)})
9	5, 8	hbfv 2837	. . . . 5 ⊢ (y ∈ (R1 ‘suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)}) → ∀x y ∈ (R1 ‘suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)}))
10	4, 9	hbel 1172	. . . 4 ⊢ (A ∈ (R1 ‘suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)}) → ∀x A ∈ (R1 ‘suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)}))
11		suceq 2288	. . . . . 6 ⊢ (x = ∩{x ∈ On∣A ∈ (R1 ‘suc x)} → suc x = suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)})
12	11	fveq2d 2836	. . . . 5 ⊢ (x = ∩{x ∈ On∣A ∈ (R1 ‘suc x)} → (R1 ‘suc x) = (R1 ‘suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)}))
13	12	eleq2d 1156	. . . 4 ⊢ (x = ∩{x ∈ On∣A ∈ (R1 ‘suc x)} → (A ∈ (R1 ‘suc x) ↔ A ∈ (R1 ‘suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)})))
14	10, 13	onminsb 2264	. . 3 ⊢ (∃x ∈ On A ∈ (R1 ‘suc x) → A ∈ (R1 ‘suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)}))
15	3, 14	ax-mp 6	. 2 ⊢ A ∈ (R1 ‘suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)})
16	1	rankval 3512	. . . 4 ⊢ (rank ‘A) = ∩{x ∈ On∣A ∈ (R1 ‘suc x)}
17		suceq 2288	. . . 4 ⊢ ((rank ‘A) = ∩{x ∈ On∣A ∈ (R1 ‘suc x)} → suc (rank ‘A) = suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)})
18	16, 17	ax-mp 6	. . 3 ⊢ suc (rank ‘A) = suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)}
19	18	fveq2i 2835	. 2 ⊢ (R1 ‘suc (rank ‘A)) = (R1 ‘suc ∩{x ∈ On∣A ∈ (R1 ‘suc x)})
20	15, 19	eleqtrr 1162	1 ⊢ A ∈ (R1 ‘suc (rank ‘A))

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

This theorem is referenced by:  rankr1lem 3517  rankel 3524  rankval3 3525  bndrank 3526  rankpw 3528  ranklon 3540

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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