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Theorem r1limg 9762
Description: Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
r1limg ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1𝐴) = 𝑥𝐴 (𝑅1𝑥))
Distinct variable group:   𝑥,𝐴

Proof of Theorem r1limg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-r1 9755 . . . . 5 𝑅1 = rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)
21dmeqi 5902 . . . 4 dom 𝑅1 = dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)
32eleq2i 2825 . . 3 (𝐴 ∈ dom 𝑅1𝐴 ∈ dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅))
4 rdglimg 8421 . . 3 ((𝐴 ∈ dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) ∧ Lim 𝐴) → (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)‘𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴))
53, 4sylanb 581 . 2 ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)‘𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴))
61fveq1i 6889 . 2 (𝑅1𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)‘𝐴)
7 r1funlim 9757 . . . . 5 (Fun 𝑅1 ∧ Lim dom 𝑅1)
87simpli 484 . . . 4 Fun 𝑅1
9 funiunfv 7243 . . . 4 (Fun 𝑅1 𝑥𝐴 (𝑅1𝑥) = (𝑅1𝐴))
108, 9ax-mp 5 . . 3 𝑥𝐴 (𝑅1𝑥) = (𝑅1𝐴)
111imaeq1i 6054 . . . 4 (𝑅1𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
1211unieqi 4920 . . 3 (𝑅1𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
1310, 12eqtri 2760 . 2 𝑥𝐴 (𝑅1𝑥) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
145, 6, 133eqtr4g 2797 1 ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1𝐴) = 𝑥𝐴 (𝑅1𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3474  c0 4321  𝒫 cpw 4601   cuni 4907   ciun 4996  cmpt 5230  dom cdm 5675  cima 5678  Lim wlim 6362  Fun wfun 6534  cfv 6540  reccrdg 8405  𝑅1cr1 9753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-r1 9755
This theorem is referenced by:  r1lim  9763  r1tr  9767  r1ordg  9769  r1pwss  9775  r1val1  9777
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