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Theorem rdglimg 8423
Description: The value of the recursive definition generator at a limit ordinal. (Contributed by NM, 16-Nov-2014.)
Assertion
Ref Expression
rdglimg ((𝐵 ∈ dom rec(𝐹, 𝐴) ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = (rec(𝐹, 𝐴) “ 𝐵))

Proof of Theorem rdglimg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2726 . 2 (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐹‘(𝑥 dom 𝑥))))) = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐹‘(𝑥 dom 𝑥)))))
2 rdgvalg 8417 . 2 (𝑦 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘𝑦) = ((𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐹‘(𝑥 dom 𝑥)))))‘(rec(𝐹, 𝐴) ↾ 𝑦)))
3 rdgseg 8420 . 2 (𝑦 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴) ↾ 𝑦) ∈ V)
4 rdgfun 8414 . . 3 Fun rec(𝐹, 𝐴)
5 funfn 6571 . . 3 (Fun rec(𝐹, 𝐴) ↔ rec(𝐹, 𝐴) Fn dom rec(𝐹, 𝐴))
64, 5mpbi 229 . 2 rec(𝐹, 𝐴) Fn dom rec(𝐹, 𝐴)
7 rdgdmlim 8415 . . 3 Lim dom rec(𝐹, 𝐴)
8 limord 6417 . . 3 (Lim dom rec(𝐹, 𝐴) → Ord dom rec(𝐹, 𝐴))
97, 8ax-mp 5 . 2 Ord dom rec(𝐹, 𝐴)
101, 2, 3, 6, 9tz7.44-3 8406 1 ((𝐵 ∈ dom rec(𝐹, 𝐴) ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = (rec(𝐹, 𝐴) “ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  c0 4317  ifcif 4523   cuni 4902  cmpt 5224  dom cdm 5669  ran crn 5670  cima 5672  Ord word 6356  Lim wlim 6358  Fun wfun 6530   Fn wfn 6531  cfv 6536  reccrdg 8407
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408
This theorem is referenced by:  rdglim  8424  r1limg  9765
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