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Theorem tfrcllemssrecs 6099
Description: Lemma for tfrcl 6111. The union of functions acceptable for tfrcl 6111 is a subset of recs. (Contributed by Jim Kingdon, 25-Mar-2022.)
Hypotheses
Ref Expression
tfrcllemssrecs.1  |-  A  =  { f  |  E. x  e.  X  (
f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) }
tfrcllemssrecs.x  |-  ( ph  ->  Ord  X )
Assertion
Ref Expression
tfrcllemssrecs  |-  ( 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)    S( x, y, f)    X( y, f)

Proof of Theorem tfrcllemssrecs
StepHypRef Expression
1 tfrcllemssrecs.1 . . . 4  |-  A  =  { f  |  E. x  e.  X  (
f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) }
2 tfrcllemssrecs.x . . . . . 6  |-  ( ph  ->  Ord  X )
3 ordsson 4299 . . . . . 6  |-  ( Ord 
X  ->  X  C_  On )
4 ssrexv 3084 . . . . . 6  |-  ( X 
C_  On  ->  ( E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) )  ->  E. x  e.  On  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) ) )
52, 3, 43syl 17 . . . . 5  |-  ( ph  ->  ( E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  (
f `  y )  =  ( G `  ( f  |`  y
) ) )  ->  E. x  e.  On  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) ) )
65ss2abdv 3092 . . . 4  |-  ( ph  ->  { f  |  E. x  e.  X  (
f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) } 
C_  { f  |  E. x  e.  On  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) } )
71, 6syl5eqss 3068 . . 3  |-  ( ph  ->  A  C_  { f  |  E. x  e.  On  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) } )
87unissd 3672 . 2  |-  ( ph  ->  U. A  C_  U. {
f  |  E. x  e.  On  ( f : x --> S  /\  A. y  e.  x  (
f `  y )  =  ( G `  ( f  |`  y
) ) ) } )
9 ffn 5147 . . . . . . 7  |-  ( f : x --> S  -> 
f  Fn  x )
109anim1i 333 . . . . . 6  |-  ( ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) )  -> 
( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( G `
 ( f  |`  y ) ) ) )
1110reximi 2470 . . . . 5  |-  ( E. x  e.  On  (
f : x --> S  /\  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 ) ) ) )
1211ss2abi 3091 . . . 4  |-  { f  |  E. x  e.  On  ( f : x --> S  /\  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 ) ) ) }
1312unissi 3671 . . 3  |-  U. {
f  |  E. x  e.  On  ( f : x --> S  /\  A. y  e.  x  (
f `  y )  =  ( G `  ( f  |`  y
) ) ) } 
C_  U. { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( G `
 ( f  |`  y ) ) ) }
14 df-recs 6052 . . 3  |- recs ( G )  =  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
1513, 14sseqtr4i 3057 . 2  |-  U. {
f  |  E. x  e.  On  ( f : x --> S  /\  A. y  e.  x  (
f `  y )  =  ( G `  ( f  |`  y
) ) ) } 
C_ recs ( G )
168, 15syl6ss 3035 1  |-  ( ph  ->  U. A  C_ recs ( G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289   {cab 2074   A.wral 2359   E.wrex 2360    C_ wss 2997   U.cuni 3648   Ord word 4180   Oncon0 4181    |` cres 4430    Fn wfn 4997   -->wf 4998   ` cfv 5002  recscrecs 6051
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3003  df-ss 3010  df-uni 3649  df-tr 3929  df-iord 4184  df-on 4186  df-f 5006  df-recs 6052
This theorem is referenced by:  tfrcllembfn  6104  tfrcllemubacc  6106  tfrcllemres  6109
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