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Theorem r1limg 8585
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 8578 . . . . 5 𝑅1 = rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)
21dmeqi 5290 . . . 4 dom 𝑅1 = dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)
32eleq2i 2690 . . 3 (𝐴 ∈ dom 𝑅1𝐴 ∈ dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅))
4 rdglimg 7473 . . 3 ((𝐴 ∈ dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) ∧ Lim 𝐴) → (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)‘𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴))
53, 4sylanb 489 . 2 ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)‘𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴))
61fveq1i 6154 . 2 (𝑅1𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)‘𝐴)
7 r1funlim 8580 . . . . 5 (Fun 𝑅1 ∧ Lim dom 𝑅1)
87simpli 474 . . . 4 Fun 𝑅1
9 funiunfv 6466 . . . 4 (Fun 𝑅1 𝑥𝐴 (𝑅1𝑥) = (𝑅1𝐴))
108, 9ax-mp 5 . . 3 𝑥𝐴 (𝑅1𝑥) = (𝑅1𝐴)
111imaeq1i 5427 . . . 4 (𝑅1𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
1211unieqi 4416 . . 3 (𝑅1𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
1310, 12eqtri 2643 . 2 𝑥𝐴 (𝑅1𝑥) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
145, 6, 133eqtr4g 2680 1 ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1𝐴) = 𝑥𝐴 (𝑅1𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3189  c0 3896  𝒫 cpw 4135   cuni 4407   ciun 4490  cmpt 4678  dom cdm 5079  cima 5082  Lim wlim 5688  Fun wfun 5846  cfv 5852  reccrdg 7457  𝑅1cr1 8576
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-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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-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-wrecs 7359  df-recs 7420  df-rdg 7458  df-r1 8578
This theorem is referenced by:  r1lim  8586  r1tr  8590  r1ordg  8592  r1pwss  8598  r1val1  8600
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