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Theorem tfr1 6408
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
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.

Proof of Theorem tfr1
StepHypRef Expression
1 eqid 2284 . . . 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 6394 . . 3  |-  Fun recs ( G )
31tfrlem14 6402 . . 3  |-  dom recs ( G )  =  On
4 df-fn 5224 . . 3  |-  (recs ( G )  Fn  On  <->  ( Fun recs ( G )  /\  dom recs ( G
)  =  On ) )
52, 3, 4mpbir2an 888 . 2  |- recs ( G )  Fn  On
6 tfr.1 . . 3  |-  F  = recs ( G )
76fneq1i 5303 . 2  |-  ( F  Fn  On  <-> recs ( G
)  Fn  On )
85, 7mpbir 202 1  |-  F  Fn  On
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1624   {cab 2270   A.wral 2544   E.wrex 2545   Oncon0 4391   dom cdm 4688    |` cres 4690   Fun wfun 5215    Fn wfn 5216   ` cfv 5221  recscrecs 6382
This theorem is referenced by:  tfr2  6409  tfr3  6410  recsfnon  6411  rdgfnon  6426  dfac8alem  7651  dfac12lem1  7764  dfac12lem2  7765  zorn2lem1  8118  zorn2lem2  8119  zorn2lem4  8121  zorn2lem5  8122  zorn2lem6  8123  zorn2lem7  8124  ttukeylem3  8133  ttukeylem5  8135  ttukeylem6  8136  dnnumch1  26540  dnnumch3lem  26542  dnnumch3  26543  aomclem6  26555
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-recs 6383
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