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Theorem rankvalb 9818
Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9837 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 ∈ βˆͺ (𝑅1 β€œ On) β†’ (rankβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})
Distinct variable group:   π‘₯,𝐴

Proof of Theorem rankvalb
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-rank 9786 . 2 rank = (𝑦 ∈ V ↦ ∩ {π‘₯ ∈ On ∣ 𝑦 ∈ (𝑅1β€˜suc π‘₯)})
2 eleq1 2813 . . . 4 (𝑦 = 𝐴 β†’ (𝑦 ∈ (𝑅1β€˜suc π‘₯) ↔ 𝐴 ∈ (𝑅1β€˜suc π‘₯)))
32rabbidv 3427 . . 3 (𝑦 = 𝐴 β†’ {π‘₯ ∈ On ∣ 𝑦 ∈ (𝑅1β€˜suc π‘₯)} = {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})
43inteqd 4947 . 2 (𝑦 = 𝐴 β†’ ∩ {π‘₯ ∈ On ∣ 𝑦 ∈ (𝑅1β€˜suc π‘₯)} = ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})
5 elex 3482 . 2 (𝐴 ∈ βˆͺ (𝑅1 β€œ On) β†’ 𝐴 ∈ V)
6 rankwflemb 9814 . . 3 (𝐴 ∈ βˆͺ (𝑅1 β€œ On) ↔ βˆƒπ‘₯ ∈ On 𝐴 ∈ (𝑅1β€˜suc π‘₯))
7 intexrab 5335 . . 3 (βˆƒπ‘₯ ∈ On 𝐴 ∈ (𝑅1β€˜suc π‘₯) ↔ ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)} ∈ V)
86, 7sylbb 218 . 2 (𝐴 ∈ βˆͺ (𝑅1 β€œ On) β†’ ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)} ∈ V)
91, 4, 5, 8fvmptd3 7021 1 (𝐴 ∈ βˆͺ (𝑅1 β€œ On) β†’ (rankβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3060  {crab 3419  Vcvv 3463  βˆͺ cuni 4901  βˆ© cint 4942   β€œ cima 5673  Oncon0 6362  suc csuc 6364  β€˜cfv 6541  π‘…1cr1 9783  rankcrnk 9784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5292  ax-nul 5299  ax-pow 5357  ax-pr 5421  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3958  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4943  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7417  df-om 7867  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-r1 9785  df-rank 9786
This theorem is referenced by:  rankr1ai  9819  rankidb  9821  rankval  9837
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