Theorem tfr1 2962

Description: Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47. We start with an arbitrary class G, normally a function, and define a class A of all "acceptable" functions. The final function we're interested in is the union F of them. F is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of F. In this first part we show that F is a function whose domain is all ordinal numbers.

Hypotheses

Ref	Expression
tfr.1	⊢ A = {f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))}
tfr.2	⊢ F = ∪A

Assertion

Ref	Expression
tfr1	⊢ F Fn On

Distinct variable group(s):   x,y,f,A   x,F,y,f   x,G,y,f

Proof of Theorem tfr1

Step	Hyp	Ref	Expression
1		tfr.1	. . . 4 ⊢ A = {f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))}
2		tfr.2	. . . 4 ⊢ F = ∪A
3	1, 2	tfrlem7 2955	. . 3 ⊢ Fun F
4		cleqid 1102	. . . 4 ⊢ (F ∪ {⟨dom F, (G ‘(F ↾ dom F))⟩}) = (F ∪ {⟨dom F, (G ‘(F ↾ dom F))⟩})
5	1, 2, 4	tfrlem13 2961	. . 3 ⊢ dom F = On
6	3, 5	pm3.2i 234	. 2 ⊢ (Fun F ∧ dom F = On)
7		df-fn 2433	. 2 ⊢ (F Fn On ↔ (Fun F ∧ dom F = On))
8	6, 7	mpbir 165	1 ⊢ F Fn On

Syntax hints:   ∧ wa 196  {cab 1090   = wceq 1091  ∀wral 1201  ∃wrex 1202   ∪ cun 1485  {csn 1808  ⟨cop 1810  ∪cuni 1919  Oncon0 2199  dom cdm 2410   ↾ cres 2412  Fun wfun 2416   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  tfr3 2964  rdgfnon 2977  numthlem 3598  zornlem2 3604  zornlem4 3606  zornlem6 3608

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

metamath.org