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Theorem r1limg 8517
 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 8510 . . . . 5 𝑅1 = rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)
21dmeqi 5247 . . . 4 dom 𝑅1 = dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)
32eleq2i 2680 . . 3 (𝐴 ∈ dom 𝑅1𝐴 ∈ dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅))
4 rdglimg 7408 . . 3 ((𝐴 ∈ dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) ∧ Lim 𝐴) → (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)‘𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴))
53, 4sylanb 488 . 2 ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)‘𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴))
61fveq1i 6104 . 2 (𝑅1𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)‘𝐴)
7 r1funlim 8512 . . . . 5 (Fun 𝑅1 ∧ Lim dom 𝑅1)
87simpli 473 . . . 4 Fun 𝑅1
9 funiunfv 6410 . . . 4 (Fun 𝑅1 𝑥𝐴 (𝑅1𝑥) = (𝑅1𝐴))
108, 9ax-mp 5 . . 3 𝑥𝐴 (𝑅1𝑥) = (𝑅1𝐴)
111imaeq1i 5382 . . . 4 (𝑅1𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
1211unieqi 4381 . . 3 (𝑅1𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
1310, 12eqtri 2632 . 2 𝑥𝐴 (𝑅1𝑥) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
145, 6, 133eqtr4g 2669 1 ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1𝐴) = 𝑥𝐴 (𝑅1𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372  ∪ ciun 4455   ↦ cmpt 4643  dom cdm 5038   “ cima 5041  Lim wlim 5641  Fun wfun 5798  ‘cfv 5804  reccrdg 7392  𝑅1cr1 8508 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510 This theorem is referenced by:  r1lim  8518  r1tr  8522  r1ordg  8524  r1pwss  8530  r1val1  8532
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