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Theorem rankidb 9416
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 9409 . . 3 (𝐴 (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
2 nfcv 2904 . . . . . 6 𝑥𝑅1
3 nfrab1 3296 . . . . . . . 8 𝑥{𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
43nfint 4869 . . . . . . 7 𝑥 {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
54nfsuc 6284 . . . . . 6 𝑥 suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
62, 5nffv 6727 . . . . 5 𝑥(𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
76nfel2 2922 . . . 4 𝑥 𝐴 ∈ (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
8 suceq 6278 . . . . . 6 (𝑥 = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → suc 𝑥 = suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
98fveq2d 6721 . . . . 5 (𝑥 = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → (𝑅1‘suc 𝑥) = (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
109eleq2d 2823 . . . 4 (𝑥 = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → (𝐴 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})))
117, 10onminsb 7578 . . 3 (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
121, 11sylbi 220 . 2 (𝐴 (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
13 rankvalb 9413 . . . 4 (𝐴 (𝑅1 “ On) → (rank‘𝐴) = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
14 suceq 6278 . . . 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 6721 . 2 (𝐴 (𝑅1 “ On) → (𝑅1‘suc (rank‘𝐴)) = (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
1712, 16eleqtrrd 2841 1 (𝐴 (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  wrex 3062  {crab 3065   cuni 4819   cint 4859  cima 5554  Oncon0 6213  suc csuc 6215  cfv 6380  𝑅1cr1 9378  rankcrnk 9379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-om 7645  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-r1 9380  df-rank 9381
This theorem is referenced by:  rankdmr1  9417  rankr1ag  9418  sswf  9424  uniwf  9435  rankonidlem  9444  rankid  9449  dfac12lem2  9758  aomclem4  40585
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