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Theorem tfr1a 6406
Description: A weak version of tfr1 6409 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
)
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.

Proof of Theorem tfr1a
StepHypRef Expression
1 eqid 2285 . . . 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 6395 . . 3  |-  Fun recs ( G )
3 tfr.1 . . . 4  |-  F  = recs ( G )
43funeqi 5242 . . 3  |-  ( Fun 
F  <->  Fun recs ( G ) )
52, 4mpbir 202 . 2  |-  Fun  F
61tfrlem16 6405 . . 3  |-  Lim  dom recs ( G )
73dmeqi 4880 . . . 4  |-  dom  F  =  dom recs ( G )
8 limeq 4404 . . . 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 1624   {cab 2271   A.wral 2545   E.wrex 2546   Oncon0 4392   Lim wlim 4393   dom cdm 4689    |` cres 4691   Fun wfun 5216    Fn wfn 5217   ` cfv 5222  recscrecs 6383
This theorem is referenced by:  tfr2b  6408  rdgfun  6425  rdgdmlim  6426  ordtypelem3  7231  ordtypelem4  7232  ordtypelem5  7233  ordtypelem6  7234  ordtypelem7  7235  ordtypelem9  7237
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-recs 6384
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