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Theorem rankvalb 8957
Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 8976 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 8925 . 2 rank = (𝑦 ∈ V ↦ {𝑥 ∈ On ∣ 𝑦 ∈ (𝑅1‘suc 𝑥)})
2 eleq1 2847 . . . 4 (𝑦 = 𝐴 → (𝑦 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ (𝑅1‘suc 𝑥)))
32rabbidv 3386 . . 3 (𝑦 = 𝐴 → {𝑥 ∈ On ∣ 𝑦 ∈ (𝑅1‘suc 𝑥)} = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
43inteqd 4715 . 2 (𝑦 = 𝐴 {𝑥 ∈ On ∣ 𝑦 ∈ (𝑅1‘suc 𝑥)} = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
5 elex 3414 . 2 (𝐴 (𝑅1 “ On) → 𝐴 ∈ V)
6 rankwflemb 8953 . . 3 (𝐴 (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
7 intexrab 5057 . . 3 (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥) ↔ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ∈ V)
86, 7sylbb 211 . 2 (𝐴 (𝑅1 “ On) → {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ∈ V)
91, 4, 5, 8fvmptd3 6564 1 (𝐴 (𝑅1 “ On) → (rank‘𝐴) = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  wrex 3091  {crab 3094  Vcvv 3398   cuni 4671   cint 4710  cima 5358  Oncon0 5976  suc csuc 5978  cfv 6135  𝑅1cr1 8922  rankcrnk 8923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-om 7344  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-r1 8924  df-rank 8925
This theorem is referenced by:  rankr1ai  8958  rankidb  8960  rankval  8976
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