ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfr2a Unicode version

Theorem tfr2a 6186
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 2117 . . . 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 6184 . . 3  |-  ( A  e.  dom recs ( G
)  ->  (recs ( G ) `  A
)  =  ( G `
 (recs ( G )  |`  A )
) )
3 tfr.1 . . . 4  |-  F  = recs ( G )
43dmeqi 4710 . . 3  |-  dom  F  =  dom recs ( G )
52, 4eleq2s 2212 . 2  |-  ( A  e.  dom  F  -> 
(recs ( G ) `
 A )  =  ( G `  (recs ( G )  |`  A ) ) )
63fveq1i 5390 . 2  |-  ( F `
 A )  =  (recs ( G ) `
 A )
73reseq1i 4785 . . 3  |-  ( F  |`  A )  =  (recs ( G )  |`  A )
87fveq2i 5392 . 2  |-  ( G `
 ( F  |`  A ) )  =  ( G `  (recs ( G )  |`  A ) )
95, 6, 83eqtr4g 2175 1  |-  ( A  e.  dom  F  -> 
( F `  A
)  =  ( G `
 ( F  |`  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   {cab 2103   A.wral 2393   E.wrex 2394   Oncon0 4255   dom cdm 4509    |` cres 4511    Fn wfn 5088   ` cfv 5093  recscrecs 6169
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-res 4521  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101  df-recs 6170
This theorem is referenced by:  tfr0  6188  tfri2d  6201  tfrcl  6229  tfri2  6231  frecsuclem  6271
  Copyright terms: Public domain W3C validator