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Theorem r1val2 8652
Description: The value of the cumulative hierarchy of sets function expressed in terms of rank. Definition 15.19 of [Monk1] p. 113. (Contributed by NM, 30-Nov-2003.)
Assertion
Ref Expression
r1val2 (𝐴 ∈ On → (𝑅1𝐴) = {𝑥 ∣ (rank‘𝑥) ∈ 𝐴})
Distinct variable group:   𝑥,𝐴

Proof of Theorem r1val2
StepHypRef Expression
1 vex 3192 . . 3 𝑥 ∈ V
21rankr1a 8651 . 2 (𝐴 ∈ On → (𝑥 ∈ (𝑅1𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
32abbi2dv 2739 1 (𝐴 ∈ On → (𝑅1𝐴) = {𝑥 ∣ (rank‘𝑥) ∈ 𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  {cab 2607  Oncon0 5687  cfv 5852  𝑅1cr1 8577  rankcrnk 8578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-reg 8449  ax-inf2 8490
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-r1 8579  df-rank 8580
This theorem is referenced by:  r1val3  8653
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