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Theorem tfr2a 6658
Description: A weak version of tfr2 6661 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 2438 . . . 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 6648 . . 3  |-  ( A  e.  dom recs ( G
)  ->  (recs ( G ) `  A
)  =  ( G `
 (recs ( G )  |`  A )
) )
3 tfr.1 . . . 4  |-  F  = recs ( G )
43dmeqi 5073 . . 3  |-  dom  F  =  dom recs ( G )
52, 4eleq2s 2530 . 2  |-  ( A  e.  dom  F  -> 
(recs ( G ) `
 A )  =  ( G `  (recs ( G )  |`  A ) ) )
63fveq1i 5731 . 2  |-  ( F `
 A )  =  (recs ( G ) `
 A )
73reseq1i 5144 . . 3  |-  ( F  |`  A )  =  (recs ( G )  |`  A )
87fveq2i 5733 . 2  |-  ( G `
 ( F  |`  A ) )  =  ( G `  (recs ( G )  |`  A ) )
95, 6, 83eqtr4g 2495 1  |-  ( A  e.  dom  F  -> 
( F `  A
)  =  ( G `
 ( F  |`  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   E.wrex 2708   Oncon0 4583   dom cdm 4880    |` cres 4882    Fn wfn 5451   ` cfv 5456  recscrecs 6634
This theorem is referenced by:  tfr2  6661  rdgvalg  6679  ordtypelem3  7491
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-fv 5464  df-recs 6635
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