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Theorem tfrlemiubacc 6049
Description: The union of  B satisfies the recursion rule (lemma for tfrlemi1 6051). (Contributed by Jim Kingdon, 22-Apr-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
tfrlemiubacc  |-  ( ph  ->  A. u  e.  x  ( U. B `  u
)  =  ( F `
 ( U. B  |`  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 tfrlemiubacc
StepHypRef Expression
1 tfrlemisucfn.1 . . . . . . . . 9  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
2 tfrlemisucfn.2 . . . . . . . . 9  |-  ( ph  ->  A. x ( Fun 
F  /\  ( F `  x )  e.  _V ) )
3 tfrlemi1.3 . . . . . . . . 9  |-  B  =  { h  |  E. z  e.  x  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `
 g ) >. } ) ) }
4 tfrlemi1.4 . . . . . . . . 9  |-  ( ph  ->  x  e.  On )
5 tfrlemi1.5 . . . . . . . . 9  |-  ( ph  ->  A. z  e.  x  E. g ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) )
61, 2, 3, 4, 5tfrlemibfn 6047 . . . . . . . 8  |-  ( ph  ->  U. B  Fn  x
)
7 fndm 5078 . . . . . . . 8  |-  ( U. B  Fn  x  ->  dom  U. B  =  x
)
86, 7syl 14 . . . . . . 7  |-  ( ph  ->  dom  U. B  =  x )
91, 2, 3, 4, 5tfrlemibacc 6045 . . . . . . . . . 10  |-  ( ph  ->  B  C_  A )
109unissd 3660 . . . . . . . . 9  |-  ( ph  ->  U. B  C_  U. A
)
111recsfval 6034 . . . . . . . . 9  |- recs ( F )  =  U. A
1210, 11syl6sseqr 3062 . . . . . . . 8  |-  ( ph  ->  U. B  C_ recs ( F ) )
13 dmss 4603 . . . . . . . 8  |-  ( U. B  C_ recs ( F )  ->  dom  U. B  C_  dom recs ( F ) )
1412, 13syl 14 . . . . . . 7  |-  ( ph  ->  dom  U. B  C_  dom recs ( F ) )
158, 14eqsstr3d 3050 . . . . . 6  |-  ( ph  ->  x  C_  dom recs ( F ) )
1615sselda 3014 . . . . 5  |-  ( (
ph  /\  w  e.  x )  ->  w  e.  dom recs ( F ) )
171tfrlem9 6038 . . . . 5  |-  ( w  e.  dom recs ( F
)  ->  (recs ( F ) `  w
)  =  ( F `
 (recs ( F )  |`  w )
) )
1816, 17syl 14 . . . 4  |-  ( (
ph  /\  w  e.  x )  ->  (recs ( F ) `  w
)  =  ( F `
 (recs ( F )  |`  w )
) )
191tfrlem7 6036 . . . . . 6  |-  Fun recs ( F )
2019a1i 9 . . . . 5  |-  ( (
ph  /\  w  e.  x )  ->  Fun recs ( F ) )
2112adantr 270 . . . . 5  |-  ( (
ph  /\  w  e.  x )  ->  U. B  C_ recs
( F ) )
228eleq2d 2154 . . . . . 6  |-  ( ph  ->  ( w  e.  dom  U. B  <->  w  e.  x
) )
2322biimpar 291 . . . . 5  |-  ( (
ph  /\  w  e.  x )  ->  w  e.  dom  U. B )
24 funssfv 5293 . . . . 5  |-  ( ( Fun recs ( F )  /\  U. B  C_ recs ( F )  /\  w  e.  dom  U. B )  ->  (recs ( F ) `  w )  =  ( U. B `  w ) )
2520, 21, 23, 24syl3anc 1172 . . . 4  |-  ( (
ph  /\  w  e.  x )  ->  (recs ( F ) `  w
)  =  ( U. B `  w )
)
26 eloni 4176 . . . . . . . . 9  |-  ( x  e.  On  ->  Ord  x )
274, 26syl 14 . . . . . . . 8  |-  ( ph  ->  Ord  x )
28 ordelss 4180 . . . . . . . 8  |-  ( ( Ord  x  /\  w  e.  x )  ->  w  C_  x )
2927, 28sylan 277 . . . . . . 7  |-  ( (
ph  /\  w  e.  x )  ->  w  C_  x )
308adantr 270 . . . . . . 7  |-  ( (
ph  /\  w  e.  x )  ->  dom  U. B  =  x )
3129, 30sseqtr4d 3052 . . . . . 6  |-  ( (
ph  /\  w  e.  x )  ->  w  C_ 
dom  U. B )
32 fun2ssres 5022 . . . . . 6  |-  ( ( Fun recs ( F )  /\  U. B  C_ recs ( F )  /\  w  C_ 
dom  U. B )  -> 
(recs ( F )  |`  w )  =  ( U. B  |`  w
) )
3320, 21, 31, 32syl3anc 1172 . . . . 5  |-  ( (
ph  /\  w  e.  x )  ->  (recs ( F )  |`  w
)  =  ( U. B  |`  w ) )
3433fveq2d 5272 . . . 4  |-  ( (
ph  /\  w  e.  x )  ->  ( F `  (recs ( F )  |`  w
) )  =  ( F `  ( U. B  |`  w ) ) )
3518, 25, 343eqtr3d 2125 . . 3  |-  ( (
ph  /\  w  e.  x )  ->  ( U. B `  w )  =  ( F `  ( U. B  |`  w
) ) )
3635ralrimiva 2442 . 2  |-  ( ph  ->  A. w  e.  x  ( U. B `  w
)  =  ( F `
 ( U. B  |`  w ) ) )
37 fveq2 5268 . . . 4  |-  ( u  =  w  ->  ( U. B `  u )  =  ( U. B `  w ) )
38 reseq2 4676 . . . . 5  |-  ( u  =  w  ->  ( U. B  |`  u )  =  ( U. B  |`  w ) )
3938fveq2d 5272 . . . 4  |-  ( u  =  w  ->  ( F `  ( U. B  |`  u ) )  =  ( F `  ( U. B  |`  w
) ) )
4037, 39eqeq12d 2099 . . 3  |-  ( u  =  w  ->  (
( U. B `  u )  =  ( F `  ( U. B  |`  u ) )  <-> 
( U. B `  w )  =  ( F `  ( U. B  |`  w ) ) ) )
4140cbvralv 2586 . 2  |-  ( A. u  e.  x  ( U. B `  u )  =  ( F `  ( U. B  |`  u
) )  <->  A. w  e.  x  ( U. B `  w )  =  ( F `  ( U. B  |`  w
) ) )
4236, 41sylibr 132 1  |-  ( ph  ->  A. u  e.  x  ( U. B `  u
)  =  ( F `
 ( U. B  |`  u ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 922   A.wal 1285    = wceq 1287   E.wex 1424    e. wcel 1436   {cab 2071   A.wral 2355   E.wrex 2356   _Vcvv 2615    u. cun 2986    C_ wss 2988   {csn 3431   <.cop 3434   U.cuni 3636   Ord word 4163   Oncon0 4164   dom cdm 4411    |` cres 4413   Fun wfun 4975    Fn wfn 4976   ` cfv 4981  recscrecs 6023
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-in1 577  ax-in2 578  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-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-id 4094  df-iord 4167  df-on 4169  df-suc 4172  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-fv 4989  df-recs 6024
This theorem is referenced by:  tfrlemiex  6050
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