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Theorem r1limg 9189
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 9182 . . . . 5 𝑅1 = rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)
21dmeqi 5772 . . . 4 dom 𝑅1 = dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)
32eleq2i 2909 . . 3 (𝐴 ∈ dom 𝑅1𝐴 ∈ dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅))
4 rdglimg 8052 . . 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 6668 . 2 (𝑅1𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)‘𝐴)
7 r1funlim 9184 . . . . 5 (Fun 𝑅1 ∧ Lim dom 𝑅1)
87simpli 484 . . . 4 Fun 𝑅1
9 funiunfv 7001 . . . 4 (Fun 𝑅1 𝑥𝐴 (𝑅1𝑥) = (𝑅1𝐴))
108, 9ax-mp 5 . . 3 𝑥𝐴 (𝑅1𝑥) = (𝑅1𝐴)
111imaeq1i 5924 . . . 4 (𝑅1𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
1211unieqi 4846 . . 3 (𝑅1𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
1310, 12eqtri 2849 . 2 𝑥𝐴 (𝑅1𝑥) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
145, 6, 133eqtr4g 2886 1 ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1𝐴) = 𝑥𝐴 (𝑅1𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  Vcvv 3500  c0 4295  𝒫 cpw 4542   cuni 4837   ciun 4917  cmpt 5143  dom cdm 5554  cima 5557  Lim wlim 6190  Fun wfun 6346  cfv 6352  reccrdg 8036  𝑅1cr1 9180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-r1 9182
This theorem is referenced by:  r1lim  9190  r1tr  9194  r1ordg  9196  r1pwss  9202  r1val1  9204
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