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Theorem r1val2 7747
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  e.  On  ->  ( R1 `  A )  =  { x  |  (
rank `  x )  e.  A } )
Distinct variable group:    x, A

Proof of Theorem r1val2
StepHypRef Expression
1 vex 2946 . . 3  |-  x  e. 
_V
21rankr1a 7746 . 2  |-  ( A  e.  On  ->  (
x  e.  ( R1
`  A )  <->  ( rank `  x )  e.  A
) )
32abbi2dv 2545 1  |-  ( A  e.  On  ->  ( R1 `  A )  =  { x  |  (
rank `  x )  e.  A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {cab 2416   Oncon0 4568   ` cfv 5440   R1cr1 7672   rankcrnk 7673
This theorem is referenced by:  r1val3  7748
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-reg 7544  ax-inf2 7580
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-recs 6619  df-rdg 6654  df-r1 7674  df-rank 7675
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