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Theorem tfr1a 7065
Description: A weak version of tfr1 7068 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 2443 . . . 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 7054 . . 3  |-  Fun recs ( G )
3 tfr.1 . . . 4  |-  F  = recs ( G )
43funeqi 5598 . . 3  |-  ( Fun 
F  <->  Fun recs ( G ) )
52, 4mpbir 209 . 2  |-  Fun  F
61tfrlem16 7064 . . 3  |-  Lim  dom recs ( G )
73dmeqi 5194 . . . 4  |-  dom  F  =  dom recs ( G )
8 limeq 4880 . . . 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 1383   {cab 2428   A.wral 2793   E.wrex 2794   Oncon0 4868   Lim wlim 4869   dom cdm 4989    |` cres 4991   Fun wfun 5572    Fn wfn 5573   ` cfv 5578  recscrecs 7043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-recs 7044
This theorem is referenced by:  tfr2b  7067  rdgfun  7084  rdgdmlim  7085  ordtypelem3  7948  ordtypelem4  7949  ordtypelem5  7950  ordtypelem6  7951  ordtypelem7  7952  ordtypelem9  7954
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