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Theorem rankval 9244
 Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). (Contributed by NM, 24-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
Hypothesis
Ref Expression
rankval.1 𝐴 ∈ V
Assertion
Ref Expression
rankval (rank‘𝐴) = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
Distinct variable group:   𝑥,𝐴

Proof of Theorem rankval
StepHypRef Expression
1 rankval.1 . . 3 𝐴 ∈ V
2 unir1 9241 . . 3 (𝑅1 “ On) = V
31, 2eleqtrri 2915 . 2 𝐴 (𝑅1 “ On)
4 rankvalb 9225 . 2 (𝐴 (𝑅1 “ On) → (rank‘𝐴) = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
53, 4ax-mp 5 1 (rank‘𝐴) = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2115  {crab 3137  Vcvv 3480  ∪ cuni 4824  ∩ cint 4862   “ cima 5546  Oncon0 6180  suc csuc 6182  ‘cfv 6345  𝑅1cr1 9190  rankcrnk 9191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457  ax-reg 9055  ax-inf2 9103 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-om 7577  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-r1 9192  df-rank 9193 This theorem is referenced by:  rankvalg  9245  rankeq1o  33717
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