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Theorem rankidb 9840
Description: Identity law for the rank function. (Contributed by NM, 3-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)
Assertion
Ref Expression
rankidb (𝐴 (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))

Proof of Theorem rankidb
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 rankwflemb 9833 . . 3 (𝐴 (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
2 nfcv 2905 . . . . . 6 𝑥𝑅1
3 nfrab1 3457 . . . . . . . 8 𝑥{𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
43nfint 4956 . . . . . . 7 𝑥 {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
54nfsuc 6456 . . . . . 6 𝑥 suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
62, 5nffv 6916 . . . . 5 𝑥(𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
76nfel2 2924 . . . 4 𝑥 𝐴 ∈ (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
8 suceq 6450 . . . . . 6 (𝑥 = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → suc 𝑥 = suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
98fveq2d 6910 . . . . 5 (𝑥 = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → (𝑅1‘suc 𝑥) = (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
109eleq2d 2827 . . . 4 (𝑥 = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → (𝐴 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})))
117, 10onminsb 7814 . . 3 (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
121, 11sylbi 217 . 2 (𝐴 (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
13 rankvalb 9837 . . . 4 (𝐴 (𝑅1 “ On) → (rank‘𝐴) = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
14 suceq 6450 . . . 4 ((rank‘𝐴) = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → suc (rank‘𝐴) = suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
1513, 14syl 17 . . 3 (𝐴 (𝑅1 “ On) → suc (rank‘𝐴) = suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
1615fveq2d 6910 . 2 (𝐴 (𝑅1 “ On) → (𝑅1‘suc (rank‘𝐴)) = (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
1712, 16eleqtrrd 2844 1 (𝐴 (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wrex 3070  {crab 3436   cuni 4907   cint 4946  cima 5688  Oncon0 6384  suc csuc 6386  cfv 6561  𝑅1cr1 9802  rankcrnk 9803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-r1 9804  df-rank 9805
This theorem is referenced by:  rankdmr1  9841  rankr1ag  9842  sswf  9848  uniwf  9859  rankonidlem  9868  rankid  9873  dfac12lem2  10185  aomclem4  43069
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