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Theorem rankvalg 9848
Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9847 expresses the class existence requirement as an antecedent instead of a hypothesis. (Contributed by NM, 5-Oct-2003.)
Assertion
Ref Expression
rankvalg (𝐴 ∈ 𝑉 β†’ (rankβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})
Distinct variable group:   π‘₯,𝐴
Allowed substitution hint:   𝑉(π‘₯)

Proof of Theorem rankvalg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6902 . . 3 (𝑦 = 𝐴 β†’ (rankβ€˜π‘¦) = (rankβ€˜π΄))
2 eleq1 2817 . . . . 5 (𝑦 = 𝐴 β†’ (𝑦 ∈ (𝑅1β€˜suc π‘₯) ↔ 𝐴 ∈ (𝑅1β€˜suc π‘₯)))
32rabbidv 3438 . . . 4 (𝑦 = 𝐴 β†’ {π‘₯ ∈ On ∣ 𝑦 ∈ (𝑅1β€˜suc π‘₯)} = {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})
43inteqd 4958 . . 3 (𝑦 = 𝐴 β†’ ∩ {π‘₯ ∈ On ∣ 𝑦 ∈ (𝑅1β€˜suc π‘₯)} = ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})
51, 4eqeq12d 2744 . 2 (𝑦 = 𝐴 β†’ ((rankβ€˜π‘¦) = ∩ {π‘₯ ∈ On ∣ 𝑦 ∈ (𝑅1β€˜suc π‘₯)} ↔ (rankβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)}))
6 vex 3477 . . 3 𝑦 ∈ V
76rankval 9847 . 2 (rankβ€˜π‘¦) = ∩ {π‘₯ ∈ On ∣ 𝑦 ∈ (𝑅1β€˜suc π‘₯)}
85, 7vtoclg 3542 1 (𝐴 ∈ 𝑉 β†’ (rankβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  {crab 3430  βˆ© cint 4953  Oncon0 6374  suc csuc 6376  β€˜cfv 6553  π‘…1cr1 9793  rankcrnk 9794
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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-reg 9623  ax-inf2 9672
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-r1 9795  df-rank 9796
This theorem is referenced by:  rankval2  9849
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