Theorem rdgfnon 2977

Description: The recursive definition generator is a function on ordinal numbers.

Assertion

Ref	Expression
rdgfnon	⊢ rec(F, A) Fn On

Proof of Theorem rdgfnon

Step	Hyp	Ref	Expression
1		rdglem1 2975	. 2 ⊢ {w∣∃u ∈ On (w Fn u ∧ ∀v ∈ u (w ‘v) = ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(w ↾ v)))} = {f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(f ↾ y)))}
2		df-rdg 2970	. 2 ⊢ rec(F, A) = ∪{w∣∃u ∈ On (w Fn u ∧ ∀v ∈ u (w ‘v) = ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(w ↾ v)))}
3	1, 2	tfr1 2962	1 ⊢ rec(F, A) Fn On

Syntax hints:  ¬ wn 1   ∨ wo 195   ∧ wa 196   ∨ w3o 580  {cab 1090   = wceq 1091  ∀wral 1201  ∃wrex 1202  ∅c0 1707  ∪cuni 1919  {copab 2055  Oncon0 2199  Lim wlim 2200  dom cdm 2410  ran crn 2411   ↾ cres 2412   Fn wfn 2417   ‘cfv 2422  reccrdg 2969

This theorem is referenced by:  rdgzer 2979  rdgsuc 2980  rdglim 2981  rdgsucopabn 2985  rdglim2 2987  frfnom 2989  abianfp 3000  r1fnon 3494  alephfnon 3668  uzrdgval 4657

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970

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