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Theorem tfr1a 6845
Description: A weak version of tfr1 6848 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 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 ) ) ) }
21tfrlem7 6834 . . 3  |-  Fun recs ( G )
3 tfr.1 . . . 4  |-  F  = recs ( G )
43funeqi 5433 . . 3  |-  ( Fun 
F  <->  Fun recs ( G ) )
52, 4mpbir 209 . 2  |-  Fun  F
61tfrlem16 6844 . . 3  |-  Lim  dom recs ( G )
73dmeqi 5036 . . . 4  |-  dom  F  =  dom recs ( G )
8 limeq 4726 . . . 4  |-  ( dom 
F  =  dom recs ( G )  ->  ( Lim  dom  F  <->  Lim  dom recs ( G ) ) )
97, 8ax-mp 5 . . 3  |-  ( Lim 
dom  F  <->  Lim  dom recs ( G
) )
106, 9mpbir 209 . 2  |-  Lim  dom  F
115, 10pm3.2i 455 1  |-  ( Fun 
F  /\  Lim  dom  F
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369   {cab 2424   A.wral 2710   E.wrex 2711   Oncon0 4714   Lim wlim 4715   dom cdm 4835    |` cres 4837   Fun wfun 5407    Fn wfn 5408   ` cfv 5413  recscrecs 6823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6824
This theorem is referenced by:  tfr2b  6847  rdgfun  6864  rdgdmlim  6865  ordtypelem3  7726  ordtypelem4  7727  ordtypelem5  7728  ordtypelem6  7729  ordtypelem7  7730  ordtypelem9  7732
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