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Theorem tfr2a 6281
Description: A weak version of transfinite recursion. (Contributed by Mario Carneiro, 24-Jun-2015.)
Hypothesis
Ref Expression
tfr.1  |-  F  = recs ( G )
Assertion
Ref Expression
tfr2a  |-  ( A  e.  dom  F  -> 
( F `  A
)  =  ( G `
 ( F  |`  A ) ) )

Proof of Theorem tfr2a
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2164 . . . 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 ) ) ) }
21tfrlem9 6279 . . 3  |-  ( A  e.  dom recs ( G
)  ->  (recs ( G ) `  A
)  =  ( G `
 (recs ( G )  |`  A )
) )
3 tfr.1 . . . 4  |-  F  = recs ( G )
43dmeqi 4800 . . 3  |-  dom  F  =  dom recs ( G )
52, 4eleq2s 2259 . 2  |-  ( A  e.  dom  F  -> 
(recs ( G ) `
 A )  =  ( G `  (recs ( G )  |`  A ) ) )
63fveq1i 5482 . 2  |-  ( F `
 A )  =  (recs ( G ) `
 A )
73reseq1i 4875 . . 3  |-  ( F  |`  A )  =  (recs ( G )  |`  A )
87fveq2i 5484 . 2  |-  ( G `
 ( F  |`  A ) )  =  ( G `  (recs ( G )  |`  A ) )
95, 6, 83eqtr4g 2222 1  |-  ( A  e.  dom  F  -> 
( F `  A
)  =  ( G `
 ( F  |`  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1342    e. wcel 2135   {cab 2150   A.wral 2442   E.wrex 2443   Oncon0 4336   dom cdm 4599    |` cres 4601    Fn wfn 5178   ` cfv 5183  recscrecs 6264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182  ax-setind 4509
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2724  df-sbc 2948  df-csb 3042  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-iun 3863  df-br 3978  df-opab 4039  df-mpt 4040  df-tr 4076  df-id 4266  df-iord 4339  df-on 4341  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-res 4611  df-iota 5148  df-fun 5185  df-fn 5186  df-fv 5191  df-recs 6265
This theorem is referenced by:  tfr0  6283  tfri2d  6296  tfrcl  6324  tfri2  6326  frecsuclem  6366
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