ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfr1onlemssrecs Unicode version

Theorem tfr1onlemssrecs 6236
Description: Lemma for tfr1on 6247. The union of functions acceptable for tfr1on 6247 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 4408 . . . . . 6  |-  ( Ord 
X  ->  X  C_  On )
4 ssrexv 3162 . . . . . 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 3170 . . . 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, 6eqsstrid 3143 . . 3  |-  ( ph  ->  A  C_  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( G `
 ( f  |`  y ) ) ) } )
87unissd 3760 . 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 6202 . 2  |- recs ( G )  =  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
108, 9sseqtrrdi 3146 1  |-  ( ph  ->  U. A  C_ recs ( G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331   {cab 2125   A.wral 2416   E.wrex 2417    C_ wss 3071   U.cuni 3736   Ord word 4284   Oncon0 4285    |` cres 4541    Fn wfn 5118   ` cfv 5123  recscrecs 6201
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027  df-iord 4288  df-on 4290  df-recs 6202
This theorem is referenced by:  tfr1onlembfn  6241  tfr1onlemubacc  6243  tfr1onlemres  6246
  Copyright terms: Public domain W3C validator