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Theorem rankvalg 9814
Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9813 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 6885 . . 3 (𝑦 = 𝐴 β†’ (rankβ€˜π‘¦) = (rankβ€˜π΄))
2 eleq1 2815 . . . . 5 (𝑦 = 𝐴 β†’ (𝑦 ∈ (𝑅1β€˜suc π‘₯) ↔ 𝐴 ∈ (𝑅1β€˜suc π‘₯)))
32rabbidv 3434 . . . 4 (𝑦 = 𝐴 β†’ {π‘₯ ∈ On ∣ 𝑦 ∈ (𝑅1β€˜suc π‘₯)} = {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})
43inteqd 4948 . . 3 (𝑦 = 𝐴 β†’ ∩ {π‘₯ ∈ On ∣ 𝑦 ∈ (𝑅1β€˜suc π‘₯)} = ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})
51, 4eqeq12d 2742 . 2 (𝑦 = 𝐴 β†’ ((rankβ€˜π‘¦) = ∩ {π‘₯ ∈ On ∣ 𝑦 ∈ (𝑅1β€˜suc π‘₯)} ↔ (rankβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)}))
6 vex 3472 . . 3 𝑦 ∈ V
76rankval 9813 . 2 (rankβ€˜π‘¦) = ∩ {π‘₯ ∈ On ∣ 𝑦 ∈ (𝑅1β€˜suc π‘₯)}
85, 7vtoclg 3537 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 3426  βˆ© cint 4943  Oncon0 6358  suc csuc 6360  β€˜cfv 6537  π‘…1cr1 9759  rankcrnk 9760
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-reg 9589  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-r1 9761  df-rank 9762
This theorem is referenced by:  rankval2  9815
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