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Theorem tfr1a 6614
Description: A weak version of tfr1 6617 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
tfr1a  |-  ( Fun 
F  /\  Lim  dom  F
)

Proof of Theorem tfr1a
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . . 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 ) ) ) }
21tfrlem7 6603 . . 3  |-  Fun recs ( G )
3 tfr.1 . . . 4  |-  F  = recs ( G )
43funeqi 5433 . . 3  |-  ( Fun 
F  <->  Fun recs ( G ) )
52, 4mpbir 201 . 2  |-  Fun  F
61tfrlem16 6613 . . 3  |-  Lim  dom recs ( G )
73dmeqi 5030 . . . 4  |-  dom  F  =  dom recs ( G )
8 limeq 4553 . . . 4  |-  ( dom 
F  =  dom recs ( G )  ->  ( Lim  dom  F  <->  Lim  dom recs ( G ) ) )
97, 8ax-mp 8 . . 3  |-  ( Lim 
dom  F  <->  Lim  dom recs ( G
) )
106, 9mpbir 201 . 2  |-  Lim  dom  F
115, 10pm3.2i 442 1  |-  ( Fun 
F  /\  Lim  dom  F
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649   {cab 2390   A.wral 2666   E.wrex 2667   Oncon0 4541   Lim wlim 4542   dom cdm 4837    |` cres 4839   Fun wfun 5407    Fn wfn 5408   ` cfv 5413  recscrecs 6591
This theorem is referenced by:  tfr2b  6616  rdgfun  6633  rdgdmlim  6634  ordtypelem3  7445  ordtypelem4  7446  ordtypelem5  7447  ordtypelem6  7448  ordtypelem7  7449  ordtypelem9  7451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6592
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