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Theorem rankvalg 9760
Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9759 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 6847 . . 3 (𝑦 = 𝐴 β†’ (rankβ€˜π‘¦) = (rankβ€˜π΄))
2 eleq1 2826 . . . . 5 (𝑦 = 𝐴 β†’ (𝑦 ∈ (𝑅1β€˜suc π‘₯) ↔ 𝐴 ∈ (𝑅1β€˜suc π‘₯)))
32rabbidv 3418 . . . 4 (𝑦 = 𝐴 β†’ {π‘₯ ∈ On ∣ 𝑦 ∈ (𝑅1β€˜suc π‘₯)} = {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})
43inteqd 4917 . . 3 (𝑦 = 𝐴 β†’ ∩ {π‘₯ ∈ On ∣ 𝑦 ∈ (𝑅1β€˜suc π‘₯)} = ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})
51, 4eqeq12d 2753 . 2 (𝑦 = 𝐴 β†’ ((rankβ€˜π‘¦) = ∩ {π‘₯ ∈ On ∣ 𝑦 ∈ (𝑅1β€˜suc π‘₯)} ↔ (rankβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)}))
6 vex 3452 . . 3 𝑦 ∈ V
76rankval 9759 . 2 (rankβ€˜π‘¦) = ∩ {π‘₯ ∈ On ∣ 𝑦 ∈ (𝑅1β€˜suc π‘₯)}
85, 7vtoclg 3528 1 (𝐴 ∈ 𝑉 β†’ (rankβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  {crab 3410  βˆ© cint 4912  Oncon0 6322  suc csuc 6324  β€˜cfv 6501  π‘…1cr1 9705  rankcrnk 9706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-reg 9535  ax-inf2 9584
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-r1 9707  df-rank 9708
This theorem is referenced by:  rankval2  9761
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