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Theorem tfr1onlemex 6212
Description: Lemma for tfr1on 6215. (Contributed by Jim Kingdon, 16-Mar-2022.)
Hypotheses
Ref Expression
tfr1on.f  |-  F  = recs ( G )
tfr1on.g  |-  ( ph  ->  Fun  G )
tfr1on.x  |-  ( ph  ->  Ord  X )
tfr1on.ex  |-  ( (
ph  /\  x  e.  X  /\  f  Fn  x
)  ->  ( G `  f )  e.  _V )
tfr1onlemsucfn.1  |-  A  =  { f  |  E. x  e.  X  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) }
tfr1onlembacc.3  |-  B  =  { h  |  E. z  e.  D  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( G `
 g ) >. } ) ) }
tfr1onlembacc.u  |-  ( (
ph  /\  x  e.  U. X )  ->  suc  x  e.  X )
tfr1onlembacc.4  |-  ( ph  ->  D  e.  X )
tfr1onlembacc.5  |-  ( ph  ->  A. z  e.  D  E. g ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) )
Assertion
Ref Expression
tfr1onlemex  |-  ( ph  ->  E. f ( f  Fn  D  /\  A. u  e.  D  (
f `  u )  =  ( G `  ( f  |`  u
) ) ) )
Distinct variable groups:    A, f, g, h, x, z    D, f, g, x    f, G, x, y    f, X, x    ph, f, g, h, x, z    y, g, z    B, f, g, h, w, z    u, B, f, w    D, h, w, z, x    u, D    h, G, z, y   
u, G, w    g, X, z    ph, w    y, w
Allowed substitution hints:    ph( y, u)    A( y, w, u)    B( x, y)    D( y)    F( x, y, z, w, u, f, g, h)    G( g)    X( y, w, u, h)

Proof of Theorem tfr1onlemex
StepHypRef Expression
1 tfr1on.f . . . 4  |-  F  = recs ( G )
2 tfr1on.g . . . 4  |-  ( ph  ->  Fun  G )
3 tfr1on.x . . . 4  |-  ( ph  ->  Ord  X )
4 tfr1on.ex . . . 4  |-  ( (
ph  /\  x  e.  X  /\  f  Fn  x
)  ->  ( G `  f )  e.  _V )
5 tfr1onlemsucfn.1 . . . 4  |-  A  =  { f  |  E. x  e.  X  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) }
6 tfr1onlembacc.3 . . . 4  |-  B  =  { h  |  E. z  e.  D  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( G `
 g ) >. } ) ) }
7 tfr1onlembacc.u . . . 4  |-  ( (
ph  /\  x  e.  U. X )  ->  suc  x  e.  X )
8 tfr1onlembacc.4 . . . 4  |-  ( ph  ->  D  e.  X )
9 tfr1onlembacc.5 . . . 4  |-  ( ph  ->  A. z  e.  D  E. g ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) )
101, 2, 3, 4, 5, 6, 7, 8, 9tfr1onlembex 6210 . . 3  |-  ( ph  ->  B  e.  _V )
11 uniexg 4331 . . 3  |-  ( B  e.  _V  ->  U. B  e.  _V )
1210, 11syl 14 . 2  |-  ( ph  ->  U. B  e.  _V )
131, 2, 3, 4, 5, 6, 7, 8, 9tfr1onlembfn 6209 . . 3  |-  ( ph  ->  U. B  Fn  D
)
141, 2, 3, 4, 5, 6, 7, 8, 9tfr1onlemubacc 6211 . . 3  |-  ( ph  ->  A. u  e.  D  ( U. B `  u
)  =  ( G `
 ( U. B  |`  u ) ) )
1513, 14jca 304 . 2  |-  ( ph  ->  ( U. B  Fn  D  /\  A. u  e.  D  ( U. B `  u )  =  ( G `  ( U. B  |`  u ) ) ) )
16 fneq1 5181 . . . 4  |-  ( f  =  U. B  -> 
( f  Fn  D  <->  U. B  Fn  D ) )
17 fveq1 5388 . . . . . 6  |-  ( f  =  U. B  -> 
( f `  u
)  =  ( U. B `  u )
)
18 reseq1 4783 . . . . . . 7  |-  ( f  =  U. B  -> 
( f  |`  u
)  =  ( U. B  |`  u ) )
1918fveq2d 5393 . . . . . 6  |-  ( f  =  U. B  -> 
( G `  (
f  |`  u ) )  =  ( G `  ( U. B  |`  u
) ) )
2017, 19eqeq12d 2132 . . . . 5  |-  ( f  =  U. B  -> 
( ( f `  u )  =  ( G `  ( f  |`  u ) )  <->  ( U. B `  u )  =  ( G `  ( U. B  |`  u
) ) ) )
2120ralbidv 2414 . . . 4  |-  ( f  =  U. B  -> 
( A. u  e.  D  ( f `  u )  =  ( G `  ( f  |`  u ) )  <->  A. u  e.  D  ( U. B `  u )  =  ( G `  ( U. B  |`  u
) ) ) )
2216, 21anbi12d 464 . . 3  |-  ( f  =  U. B  -> 
( ( f  Fn  D  /\  A. u  e.  D  ( f `  u )  =  ( G `  ( f  |`  u ) ) )  <-> 
( U. B  Fn  D  /\  A. u  e.  D  ( U. B `  u )  =  ( G `  ( U. B  |`  u ) ) ) ) )
2322spcegv 2748 . 2  |-  ( U. B  e.  _V  ->  ( ( U. B  Fn  D  /\  A. u  e.  D  ( U. B `  u )  =  ( G `  ( U. B  |`  u ) ) )  ->  E. f
( f  Fn  D  /\  A. u  e.  D  ( f `  u
)  =  ( G `
 ( f  |`  u ) ) ) ) )
2412, 15, 23sylc 62 1  |-  ( ph  ->  E. f ( f  Fn  D  /\  A. u  e.  D  (
f `  u )  =  ( G `  ( f  |`  u
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 947    = wceq 1316   E.wex 1453    e. wcel 1465   {cab 2103   A.wral 2393   E.wrex 2394   _Vcvv 2660    u. cun 3039   {csn 3497   <.cop 3500   U.cuni 3706   Ord word 4254   suc csuc 4257    |` cres 4511   Fun wfun 5087    Fn wfn 5088   ` cfv 5093  recscrecs 6169
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-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-recs 6170
This theorem is referenced by:  tfr1onlemaccex  6213
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