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Theorem tfr1 6842
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  = recs ( G ) 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. (Contributed by NM, 17-Aug-1994.) (Revised by Mario Carneiro, 18-Jan-2015.)
Hypothesis
Ref Expression
tfr.1  |-  F  = recs ( G )
Assertion
Ref Expression
tfr1  |-  F  Fn  On

Proof of Theorem tfr1
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2433 . . . 4  |-  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
21tfrlem7 6828 . . 3  |-  Fun recs ( G )
31tfrlem14 6836 . . 3  |-  dom recs ( G )  =  On
4 df-fn 5409 . . 3  |-  (recs ( G )  Fn  On  <->  ( Fun recs ( G )  /\  dom recs ( G
)  =  On ) )
52, 3, 4mpbir2an 904 . 2  |- recs ( G )  Fn  On
6 tfr.1 . . 3  |-  F  = recs ( G )
76fneq1i 5493 . 2  |-  ( F  Fn  On  <-> recs ( G
)  Fn  On )
85, 7mpbir 209 1  |-  F  Fn  On
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1362   {cab 2419   A.wral 2705   E.wrex 2706   Oncon0 4706   dom cdm 4827    |` cres 4829   Fun wfun 5400    Fn wfn 5401   ` cfv 5406  recscrecs 6817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-recs 6818
This theorem is referenced by:  tfr2  6843  tfr3  6844  recsfnon  6845  rdgfnon  6860  dfac8alem  8187  dfac12lem1  8300  dfac12lem2  8301  zorn2lem1  8653  zorn2lem2  8654  zorn2lem4  8656  zorn2lem5  8657  zorn2lem6  8658  zorn2lem7  8659  ttukeylem3  8668  ttukeylem5  8670  ttukeylem6  8671  dnnumch1  29242  dnnumch3lem  29244  dnnumch3  29245  aomclem6  29257
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