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Theorem tfr1a 6364
Description: A weak version of tfr1 6367 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
StepHypRef Expression
1 eqid 2256 . . . 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 6353 . . 3  |-  Fun recs ( G )
3 tfr.1 . . . 4  |-  F  = recs ( G )
43funeqi 5200 . . 3  |-  ( Fun 
F  <->  Fun recs ( G ) )
52, 4mpbir 202 . 2  |-  Fun  F
61tfrlem16 6363 . . 3  |-  Lim  dom recs ( G )
73dmeqi 4854 . . . 4  |-  dom  F  =  dom recs ( G )
8 limeq 4362 . . . 4  |-  ( dom 
F  =  dom recs ( G )  ->  ( Lim  dom  F  <->  Lim  dom recs ( G ) ) )
97, 8ax-mp 10 . . 3  |-  ( Lim 
dom  F  <->  Lim  dom recs ( G
) )
106, 9mpbir 202 . 2  |-  Lim  dom  F
115, 10pm3.2i 443 1  |-  ( Fun 
F  /\  Lim  dom  F
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1619   {cab 2242   A.wral 2516   E.wrex 2517   Oncon0 4350   Lim wlim 4351   dom cdm 4647    |` cres 4649   Fun wfun 4653    Fn wfn 4654   ` cfv 4659  recscrecs 6341
This theorem is referenced by:  tfr2b  6366  rdgfun  6383  rdgdmlim  6384  ordtypelem3  7189  ordtypelem4  7190  ordtypelem5  7191  ordtypelem6  7192  ordtypelem7  7193  ordtypelem9  7195
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-recs 6342
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