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Theorem rankidb 8701
 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 8694 . . 3 (𝐴 (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
2 nfcv 2793 . . . . . 6 𝑥𝑅1
3 nfrab1 3152 . . . . . . . 8 𝑥{𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
43nfint 4518 . . . . . . 7 𝑥 {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
54nfsuc 5834 . . . . . 6 𝑥 suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
62, 5nffv 6236 . . . . 5 𝑥(𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
76nfel2 2810 . . . 4 𝑥 𝐴 ∈ (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
8 suceq 5828 . . . . . 6 (𝑥 = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → suc 𝑥 = suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
98fveq2d 6233 . . . . 5 (𝑥 = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → (𝑅1‘suc 𝑥) = (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
109eleq2d 2716 . . . 4 (𝑥 = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → (𝐴 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})))
117, 10onminsb 7041 . . 3 (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
121, 11sylbi 207 . 2 (𝐴 (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
13 rankvalb 8698 . . . 4 (𝐴 (𝑅1 “ On) → (rank‘𝐴) = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
14 suceq 5828 . . . 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 6233 . 2 (𝐴 (𝑅1 “ On) → (𝑅1‘suc (rank‘𝐴)) = (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
1712, 16eleqtrrd 2733 1 (𝐴 (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  ∃wrex 2942  {crab 2945  ∪ cuni 4468  ∩ cint 4507   “ cima 5146  Oncon0 5761  suc csuc 5763  ‘cfv 5926  𝑅1cr1 8663  rankcrnk 8664 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-r1 8665  df-rank 8666 This theorem is referenced by:  rankdmr1  8702  rankr1ag  8703  sswf  8709  uniwf  8720  rankonidlem  8729  rankid  8734  dfac12lem2  9004  aomclem4  37944
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