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Theorem rankvalb 9223
Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9242 does not use Regularity, and so requires the assumption that 𝐴 is in the range of 𝑅1. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
Assertion
Ref Expression
rankvalb (𝐴 (𝑅1 “ On) → (rank‘𝐴) = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
Distinct variable group:   𝑥,𝐴

Proof of Theorem rankvalb
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-rank 9191 . 2 rank = (𝑦 ∈ V ↦ {𝑥 ∈ On ∣ 𝑦 ∈ (𝑅1‘suc 𝑥)})
2 eleq1 2903 . . . 4 (𝑦 = 𝐴 → (𝑦 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ (𝑅1‘suc 𝑥)))
32rabbidv 3465 . . 3 (𝑦 = 𝐴 → {𝑥 ∈ On ∣ 𝑦 ∈ (𝑅1‘suc 𝑥)} = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
43inteqd 4867 . 2 (𝑦 = 𝐴 {𝑥 ∈ On ∣ 𝑦 ∈ (𝑅1‘suc 𝑥)} = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
5 elex 3498 . 2 (𝐴 (𝑅1 “ On) → 𝐴 ∈ V)
6 rankwflemb 9219 . . 3 (𝐴 (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
7 intexrab 5229 . . 3 (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥) ↔ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ∈ V)
86, 7sylbb 222 . 2 (𝐴 (𝑅1 “ On) → {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ∈ V)
91, 4, 5, 8fvmptd3 6782 1 (𝐴 (𝑅1 “ On) → (rank‘𝐴) = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  wrex 3134  {crab 3137  Vcvv 3480   cuni 4824   cint 4862  cima 5545  Oncon0 6178  suc csuc 6180  cfv 6343  𝑅1cr1 9188  rankcrnk 9189
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-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-r1 9190  df-rank 9191
This theorem is referenced by:  rankr1ai  9224  rankidb  9226  rankval  9242
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