Theorem tfr1 3930

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 | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
tfr.2	|- F = U.A

Assertion

Ref	Expression
tfr1	|- F Fn On

Distinct variable groups:   x,y,f,A   x,F,y,f   x,G,y,f

Proof of Theorem tfr1

Step	Hyp	Ref	Expression
1		df-fn 3199	. 2 |- (F Fn On <-> (Fun F /\ dom F = On))
2		tfr.1	. . 3 |- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
3		tfr.2	. . 3 |- F = U.A
4	2, 3	tfrlem7 3923	. 2 |- Fun F
5		eqid 1478	. . 3 |- (F u. {<.dom F, (G` (F |` dom F))>.}) = (F u. {<.dom F, (G` (F |` dom F))>.})
6	2, 3, 5	tfrlem13 3929	. 2 |- dom F = On
7	1, 4, 6	mpbir2an 732	1 |- F Fn On

Syntax hints:   /\ wa 223   = wceq 958  {cab 1466  A.wral 1648  E.wrex 1649   u. cun 2048  {csn 2413  <.cop 2415  U.cuni 2507  Oncon0 2954  dom cdm 3176   |` cres 3178  Fun wfun 3182   Fn wfn 3183  ` cfv 3188

This theorem is referenced by:  tfr3 3932  rdgfnon 3945  numthlem 4793  zorn2lem2 4799  zorn2lem4 4801  zorn2lem6 4803

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-suc 2960  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204

Copyright terms: Public domain