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Theorem tfrcllemssrecs 6328
Description: Lemma for tfrcl 6340. The union of functions acceptable for tfrcl 6340 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 4474 . . . . . 6  |-  ( Ord 
X  ->  X  C_  On )
4 ssrexv 3212 . . . . . 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 3220 . . . 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, 6eqsstrid 3193 . . 3  |-  ( ph  ->  A  C_  { f  |  E. x  e.  On  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) } )
87unissd 3818 . 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 5345 . . . . . . 7  |-  ( f : x --> S  -> 
f  Fn  x )
109anim1i 338 . . . . . 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 2567 . . . . 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 3219 . . . 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 3817 . . 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 6281 . . 3  |- recs ( G )  =  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
1513, 14sseqtrri 3182 . 2  |-  U. {
f  |  E. x  e.  On  ( f : x --> S  /\  A. y  e.  x  (
f `  y )  =  ( G `  ( f  |`  y
) ) ) } 
C_ recs ( G )
168, 15sstrdi 3159 1  |-  ( ph  ->  U. A  C_ recs ( G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348   {cab 2156   A.wral 2448   E.wrex 2449    C_ wss 3121   U.cuni 3794   Ord word 4345   Oncon0 4346    |` cres 4611    Fn wfn 5191   -->wf 5192   ` cfv 5196  recscrecs 6280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3795  df-tr 4086  df-iord 4349  df-on 4351  df-f 5200  df-recs 6281
This theorem is referenced by:  tfrcllembfn  6333  tfrcllemubacc  6335  tfrcllemres  6338
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