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Theorem frectfr 6070
Description: Lemma to connect transfinite recursion theorems with finite recursion. That is, given the conditions  F  Fn  _V and  A  e.  V on frec ( F ,  A ), we want to be able to apply tfri1d 6005 or tfri2d 6006, and this lemma lets us satisfy hypotheses of those theorems.

(Contributed by Jim Kingdon, 15-Aug-2019.)

Hypothesis
Ref Expression
frectfr.1  |-  G  =  ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )
Assertion
Ref Expression
frectfr  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  ->  A. y
( Fun  G  /\  ( G `  y )  e.  _V ) )
Distinct variable groups:    g, m, x, y, A    z, g, F, m, x, y    g, V, m, y
Allowed substitution hints:    A( z)    G( x, y, z, g, m)    V( x, z)

Proof of Theorem frectfr
StepHypRef Expression
1 vex 2613 . . . . . . . 8  |-  g  e. 
_V
21a1i 9 . . . . . . 7  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  ->  g  e.  _V )
3 simpl 107 . . . . . . 7  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  ->  A. z
( F `  z
)  e.  _V )
4 simpr 108 . . . . . . 7  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  ->  A  e.  V )
52, 3, 4frecabex 6068 . . . . . 6  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  ->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) }  e.  _V )
65ralrimivw 2440 . . . . 5  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  ->  A. g  e.  _V  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) }  e.  _V )
7 frectfr.1 . . . . . 6  |-  G  =  ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )
87fnmpt 5076 . . . . 5  |-  ( A. g  e.  _V  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) }  e.  _V  ->  G  Fn  _V )
96, 8syl 14 . . . 4  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  ->  G  Fn  _V )
10 vex 2613 . . . 4  |-  y  e. 
_V
11 funfvex 5244 . . . . 5  |-  ( ( Fun  G  /\  y  e.  dom  G )  -> 
( G `  y
)  e.  _V )
1211funfni 5050 . . . 4  |-  ( ( G  Fn  _V  /\  y  e.  _V )  ->  ( G `  y
)  e.  _V )
139, 10, 12sylancl 404 . . 3  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  ->  ( G `  y )  e.  _V )
147funmpt2 4989 . . 3  |-  Fun  G
1513, 14jctil 305 . 2  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  ->  ( Fun  G  /\  ( G `
 y )  e. 
_V ) )
1615alrimiv 1797 1  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  ->  A. y
( Fun  G  /\  ( G `  y )  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 662   A.wal 1283    = wceq 1285    e. wcel 1434   {cab 2069   A.wral 2353   E.wrex 2354   _Vcvv 2610   (/)c0 3267    |-> cmpt 3859   suc csuc 4148   omcom 4359   dom cdm 4391   Fun wfun 4946    Fn wfn 4947   ` cfv 4952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960
This theorem is referenced by:  frecfnom  6071
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