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Theorem tfr2a 5936
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 2040 . . . 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 5935 . . 3  |-  ( A  e.  dom recs ( G
)  ->  (recs ( G ) `  A
)  =  ( G `
 (recs ( G )  |`  A )
) )
3 tfr.1 . . . 4  |-  F  = recs ( G )
43dmeqi 4536 . . 3  |-  dom  F  =  dom recs ( G )
52, 4eleq2s 2132 . 2  |-  ( A  e.  dom  F  -> 
(recs ( G ) `
 A )  =  ( G `  (recs ( G )  |`  A ) ) )
63fveq1i 5179 . 2  |-  ( F `
 A )  =  (recs ( G ) `
 A )
73reseq1i 4608 . . 3  |-  ( F  |`  A )  =  (recs ( G )  |`  A )
87fveq2i 5181 . 2  |-  ( G `
 ( F  |`  A ) )  =  ( G `  (recs ( G )  |`  A ) )
95, 6, 83eqtr4g 2097 1  |-  ( A  e.  dom  F  -> 
( F `  A
)  =  ( G `
 ( F  |`  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   {cab 2026   A.wral 2306   E.wrex 2307   Oncon0 4100   dom cdm 4345    |` cres 4347    Fn wfn 4897   ` cfv 4902  recscrecs 5919
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-recs 5920
This theorem is referenced by:  tfr0  5937  tfri2d  5950  tfri2  5952
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