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Theorem rankvalg 8535
Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 8534 expresses the class existence requirement as an antecedent instead of a hypothesis. (Contributed by NM, 5-Oct-2003.)
Assertion
Ref Expression
rankvalg (𝐴𝑉 → (rank‘𝐴) = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem rankvalg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6083 . . 3 (𝑦 = 𝐴 → (rank‘𝑦) = (rank‘𝐴))
2 eleq1 2670 . . . . 5 (𝑦 = 𝐴 → (𝑦 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ (𝑅1‘suc 𝑥)))
32rabbidv 3158 . . . 4 (𝑦 = 𝐴 → {𝑥 ∈ On ∣ 𝑦 ∈ (𝑅1‘suc 𝑥)} = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
43inteqd 4404 . . 3 (𝑦 = 𝐴 {𝑥 ∈ On ∣ 𝑦 ∈ (𝑅1‘suc 𝑥)} = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
51, 4eqeq12d 2619 . 2 (𝑦 = 𝐴 → ((rank‘𝑦) = {𝑥 ∈ On ∣ 𝑦 ∈ (𝑅1‘suc 𝑥)} ↔ (rank‘𝐴) = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
6 vex 3170 . . 3 𝑦 ∈ V
76rankval 8534 . 2 (rank‘𝑦) = {𝑥 ∈ On ∣ 𝑦 ∈ (𝑅1‘suc 𝑥)}
85, 7vtoclg 3233 1 (𝐴𝑉 → (rank‘𝐴) = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  {crab 2894   cint 4399  Oncon0 5621  suc csuc 5623  cfv 5785  𝑅1cr1 8480  rankcrnk 8481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-reg 8352  ax-inf2 8393
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-om 6930  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-r1 8482  df-rank 8483
This theorem is referenced by:  rankval2  8536
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