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Theorem tfr1onlemssrecs 6166
Description: Lemma for tfr1on 6177. The union of functions acceptable for tfr1on 6177 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 4346 . . . . . 6  |-  ( Ord 
X  ->  X  C_  On )
4 ssrexv 3109 . . . . . 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 3117 . . . 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 3093 . . 3  |-  ( ph  ->  A  C_  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( G `
 ( f  |`  y ) ) ) } )
87unissd 3707 . 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 6132 . 2  |- recs ( G )  =  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
108, 9syl6sseqr 3096 1  |-  ( ph  ->  U. A  C_ recs ( G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1299   {cab 2086   A.wral 2375   E.wrex 2376    C_ wss 3021   U.cuni 3683   Ord word 4222   Oncon0 4223    |` cres 4479    Fn wfn 5054   ` cfv 5059  recscrecs 6131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-in 3027  df-ss 3034  df-uni 3684  df-tr 3967  df-iord 4226  df-on 4228  df-recs 6132
This theorem is referenced by:  tfr1onlembfn  6171  tfr1onlemubacc  6173  tfr1onlemres  6176
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