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

Theorem tfrlemiex 6335
Description: Lemma for tfrlemi1 6336. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
Hypotheses
Ref Expression
tfrlemisucfn.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
tfrlemisucfn.2  |-  ( ph  ->  A. x ( Fun 
F  /\  ( F `  x )  e.  _V ) )
tfrlemi1.3  |-  B  =  { h  |  E. z  e.  x  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `
 g ) >. } ) ) }
tfrlemi1.4  |-  ( ph  ->  x  e.  On )
tfrlemi1.5  |-  ( ph  ->  A. z  e.  x  E. g ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) )
Assertion
Ref Expression
tfrlemiex  |-  ( ph  ->  E. f ( f  Fn  x  /\  A. u  e.  x  (
f `  u )  =  ( F `  ( f  |`  u
) ) ) )
Distinct variable groups:    f, g, h, u, w, x, y, z, A    f, F, g, h, u, w, x, y, z    ph, w, y    u, B, w, f, g, h, z    ph, g, h, z
Allowed substitution hints:    ph( x, u, f)    B( x, y)

Proof of Theorem tfrlemiex
StepHypRef Expression
1 tfrlemisucfn.1 . . . 4  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
2 tfrlemisucfn.2 . . . 4  |-  ( ph  ->  A. x ( Fun 
F  /\  ( F `  x )  e.  _V ) )
3 tfrlemi1.3 . . . 4  |-  B  =  { h  |  E. z  e.  x  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `
 g ) >. } ) ) }
4 tfrlemi1.4 . . . 4  |-  ( ph  ->  x  e.  On )
5 tfrlemi1.5 . . . 4  |-  ( ph  ->  A. z  e.  x  E. g ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) )
61, 2, 3, 4, 5tfrlemibex 6333 . . 3  |-  ( ph  ->  B  e.  _V )
7 uniexg 4441 . . 3  |-  ( B  e.  _V  ->  U. B  e.  _V )
86, 7syl 14 . 2  |-  ( ph  ->  U. B  e.  _V )
91, 2, 3, 4, 5tfrlemibfn 6332 . . 3  |-  ( ph  ->  U. B  Fn  x
)
101, 2, 3, 4, 5tfrlemiubacc 6334 . . 3  |-  ( ph  ->  A. u  e.  x  ( U. B `  u
)  =  ( F `
 ( U. B  |`  u ) ) )
119, 10jca 306 . 2  |-  ( ph  ->  ( U. B  Fn  x  /\  A. u  e.  x  ( U. B `  u )  =  ( F `  ( U. B  |`  u ) ) ) )
12 fneq1 5306 . . . 4  |-  ( f  =  U. B  -> 
( f  Fn  x  <->  U. B  Fn  x ) )
13 fveq1 5516 . . . . . 6  |-  ( f  =  U. B  -> 
( f `  u
)  =  ( U. B `  u )
)
14 reseq1 4903 . . . . . . 7  |-  ( f  =  U. B  -> 
( f  |`  u
)  =  ( U. B  |`  u ) )
1514fveq2d 5521 . . . . . 6  |-  ( f  =  U. B  -> 
( F `  (
f  |`  u ) )  =  ( F `  ( U. B  |`  u
) ) )
1613, 15eqeq12d 2192 . . . . 5  |-  ( f  =  U. B  -> 
( ( f `  u )  =  ( F `  ( f  |`  u ) )  <->  ( U. B `  u )  =  ( F `  ( U. B  |`  u
) ) ) )
1716ralbidv 2477 . . . 4  |-  ( f  =  U. B  -> 
( A. u  e.  x  ( f `  u )  =  ( F `  ( f  |`  u ) )  <->  A. u  e.  x  ( U. B `  u )  =  ( F `  ( U. B  |`  u
) ) ) )
1812, 17anbi12d 473 . . 3  |-  ( f  =  U. B  -> 
( ( f  Fn  x  /\  A. u  e.  x  ( f `  u )  =  ( F `  ( f  |`  u ) ) )  <-> 
( U. B  Fn  x  /\  A. u  e.  x  ( U. B `  u )  =  ( F `  ( U. B  |`  u ) ) ) ) )
1918spcegv 2827 . 2  |-  ( U. B  e.  _V  ->  ( ( U. B  Fn  x  /\  A. u  e.  x  ( U. B `  u )  =  ( F `  ( U. B  |`  u ) ) )  ->  E. f
( f  Fn  x  /\  A. u  e.  x  ( f `  u
)  =  ( F `
 ( f  |`  u ) ) ) ) )
208, 11, 19sylc 62 1  |-  ( ph  ->  E. f ( f  Fn  x  /\  A. u  e.  x  (
f `  u )  =  ( F `  ( f  |`  u
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978   A.wal 1351    = wceq 1353   E.wex 1492    e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   _Vcvv 2739    u. cun 3129   {csn 3594   <.cop 3597   U.cuni 3811   Oncon0 4365    |` cres 4630   Fun wfun 5212    Fn wfn 5213   ` cfv 5218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-recs 6309
This theorem is referenced by:  tfrlemi1  6336
  Copyright terms: Public domain W3C validator