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Theorem tfr1onlemssrecs 6008
Description: Lemma for tfr1on 6019. The union of functions acceptable for tfr1on 6019 is a subset of recs. (Contributed by Jim Kingdon, 15-Mar-2022.)
Hypotheses
Ref Expression
tfr1onlemssrecs.1  |-  A  =  { f  |  E. x  e.  X  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) }
tfr1onlemssrecs.x  |-  ( ph  ->  Ord  X )
Assertion
Ref Expression
tfr1onlemssrecs  |-  ( ph  ->  U. A  C_ recs ( G ) )
Distinct variable groups:    f, G, x, y    x, X    ph, f
Allowed substitution hints:    ph( x, y)    A( x, y, f)    X( y, f)

Proof of Theorem tfr1onlemssrecs
StepHypRef Expression
1 tfr1onlemssrecs.1 . . . 4  |-  A  =  { f  |  E. x  e.  X  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) }
2 tfr1onlemssrecs.x . . . . . 6  |-  ( ph  ->  Ord  X )
3 ordsson 4264 . . . . . 6  |-  ( Ord 
X  ->  X  C_  On )
4 ssrexv 3068 . . . . . 6  |-  ( X 
C_  On  ->  ( E. x  e.  X  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) )  ->  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( G `
 ( f  |`  y ) ) ) ) )
52, 3, 43syl 17 . . . . 5  |-  ( ph  ->  ( E. x  e.  X  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) )  ->  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( G `
 ( f  |`  y ) ) ) ) )
65ss2abdv 3076 . . . 4  |-  ( ph  ->  { f  |  E. x  e.  X  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) } 
C_  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( G `
 ( f  |`  y ) ) ) } )
71, 6syl5eqss 3052 . . 3  |-  ( ph  ->  A  C_  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( G `
 ( f  |`  y ) ) ) } )
87unissd 3645 . 2  |-  ( ph  ->  U. A  C_  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) } )
9 df-recs 5974 . 2  |- recs ( G )  =  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
108, 9syl6sseqr 3055 1  |-  ( ph  ->  U. A  C_ recs ( G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285   {cab 2069   A.wral 2353   E.wrex 2354    C_ wss 2982   U.cuni 3621   Ord word 4145   Oncon0 4146    |` cres 4393    Fn wfn 4947   ` cfv 4952  recscrecs 5973
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-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-in 2988  df-ss 2995  df-uni 3622  df-tr 3896  df-iord 4149  df-on 4151  df-recs 5974
This theorem is referenced by:  tfr1onlembfn  6013  tfr1onlemubacc  6015  tfr1onlemres  6018
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