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Theorem r1lim 8808
 Description: Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
r1lim ((𝐴𝐵 ∧ Lim 𝐴) → (𝑅1𝐴) = 𝑥𝐴 (𝑅1𝑥))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem r1lim
StepHypRef Expression
1 limelon 5949 . . 3 ((𝐴𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)
2 r1fnon 8803 . . . 4 𝑅1 Fn On
3 fndm 6151 . . . 4 (𝑅1 Fn On → dom 𝑅1 = On)
42, 3ax-mp 5 . . 3 dom 𝑅1 = On
51, 4syl6eleqr 2850 . 2 ((𝐴𝐵 ∧ Lim 𝐴) → 𝐴 ∈ dom 𝑅1)
6 r1limg 8807 . 2 ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1𝐴) = 𝑥𝐴 (𝑅1𝑥))
75, 6sylancom 704 1 ((𝐴𝐵 ∧ Lim 𝐴) → (𝑅1𝐴) = 𝑥𝐴 (𝑅1𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∪ ciun 4672  dom cdm 5266  Oncon0 5884  Lim wlim 5885   Fn wfn 6044  ‘cfv 6049  𝑅1cr1 8798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-r1 8800 This theorem is referenced by:  r1sdom  8810  r1om  9258  inar1  9789  inatsk  9792  grur1a  9833
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