ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfrcllemssrecs Unicode version

Theorem tfrcllemssrecs 6052
Description: Lemma for tfrcl 6064. The union of functions acceptable for tfrcl 6064 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 4275 . . . . . 6  |-  ( Ord 
X  ->  X  C_  On )
4 ssrexv 3072 . . . . . 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 3080 . . . 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 3056 . . 3  |-  ( ph  ->  A  C_  { f  |  E. x  e.  On  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) } )
87unissd 3654 . 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 5117 . . . . . . 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 2466 . . . . 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 3079 . . . 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 3653 . . 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 6005 . . 3  |- recs ( G )  =  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
1513, 14sseqtr4i 3045 . 2  |-  U. {
f  |  E. x  e.  On  ( f : x --> S  /\  A. y  e.  x  (
f `  y )  =  ( G `  ( f  |`  y
) ) ) } 
C_ recs ( G )
168, 15syl6ss 3024 1  |-  ( ph  ->  U. A  C_ recs ( G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1287   {cab 2071   A.wral 2355   E.wrex 2356    C_ wss 2986   U.cuni 3630   Ord word 4156   Oncon0 4157    |` cres 4406    Fn wfn 4967   -->wf 4968   ` cfv 4972  recscrecs 6004
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-in 2992  df-ss 2999  df-uni 3631  df-tr 3905  df-iord 4160  df-on 4162  df-f 4976  df-recs 6005
This theorem is referenced by:  tfrcllembfn  6057  tfrcllemubacc  6059  tfrcllemres  6062
  Copyright terms: Public domain W3C validator