Theorem tfr1 4225

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 3274	. 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 4218	. 2 |- Fun F
5		eqid 1518	. . 3 |- (F u. {<.dom F, (G` (F |` dom F))>.}) = (F u. {<.dom F, (G` (F |` dom F))>.})
6	2, 3, 5	tfrlem13 4224	. 2 |- dom F = On
7	1, 4, 6	mpbir2an 735	1 |- F Fn On

Syntax hints:   /\ wa 221   = wceq 992  {cab 1505  A.wral 1691  E.wrex 1692   u. cun 2097  {csn 2467  <.cop 2469  U.cuni 2569  Oncon0 2975  dom cdm 3251   |` cres 3253  Fun wfun 3257   Fn wfn 3258  ` cfv 3263

This theorem is referenced by:  tfr3 4227  rdgfnon 4240  numthlem 4929  zorn2lem2 4935  zorn2lem4 4937  zorn2lem6 4939  ordtypelem2 11428  ordtypelem4 11430  ordtypelem5 11431  ordtypelem6 11432  ordtypelem7 11433

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-rab 1698  df-v 1858  df-sbc 1987  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-suc 2981  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-fv 3279

Copyright terms: Public domain