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Theorem tfr2a 7076
Description: A weak version of tfr2 7079 which is useful for proofs that avoid the Axiom of Replacement. (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 2467 . . . 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 7066 . . 3  |-  ( A  e.  dom recs ( G
)  ->  (recs ( G ) `  A
)  =  ( G `
 (recs ( G )  |`  A )
) )
3 tfr.1 . . . 4  |-  F  = recs ( G )
43dmeqi 5210 . . 3  |-  dom  F  =  dom recs ( G )
52, 4eleq2s 2575 . 2  |-  ( A  e.  dom  F  -> 
(recs ( G ) `
 A )  =  ( G `  (recs ( G )  |`  A ) ) )
63fveq1i 5873 . 2  |-  ( F `
 A )  =  (recs ( G ) `
 A )
73reseq1i 5275 . . 3  |-  ( F  |`  A )  =  (recs ( G )  |`  A )
87fveq2i 5875 . 2  |-  ( G `
 ( F  |`  A ) )  =  ( G `  (recs ( G )  |`  A ) )
95, 6, 83eqtr4g 2533 1  |-  ( A  e.  dom  F  -> 
( F `  A
)  =  ( G `
 ( F  |`  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2817   E.wrex 2818   Oncon0 4884   dom cdm 5005    |` cres 5007    Fn wfn 5589   ` cfv 5594  recscrecs 7053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-recs 7054
This theorem is referenced by:  tfr2  7079  rdgvalg  7097  ordtypelem3  7957
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