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Theorem tfrlemiubacc 6350
Description: The union of  B satisfies the recursion rule (lemma for tfrlemi1 6352). (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 6348 . . . . . . . 8  |-  ( ph  ->  U. B  Fn  x
)
7 fndm 5331 . . . . . . . 8  |-  ( U. B  Fn  x  ->  dom  U. B  =  x
)
86, 7syl 14 . . . . . . 7  |-  ( ph  ->  dom  U. B  =  x )
91, 2, 3, 4, 5tfrlemibacc 6346 . . . . . . . . . 10  |-  ( ph  ->  B  C_  A )
109unissd 3848 . . . . . . . . 9  |-  ( ph  ->  U. B  C_  U. A
)
111recsfval 6335 . . . . . . . . 9  |- recs ( F )  =  U. A
1210, 11sseqtrrdi 3219 . . . . . . . 8  |-  ( ph  ->  U. B  C_ recs ( F ) )
13 dmss 4841 . . . . . . . 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, 14eqsstrrd 3207 . . . . . 6  |-  ( ph  ->  x  C_  dom recs ( F ) )
1615sselda 3170 . . . . 5  |-  ( (
ph  /\  w  e.  x )  ->  w  e.  dom recs ( F ) )
171tfrlem9 6339 . . . . 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 6337 . . . . . 6  |-  Fun recs ( F )
2019a1i 9 . . . . 5  |-  ( (
ph  /\  w  e.  x )  ->  Fun recs ( F ) )
2112adantr 276 . . . . 5  |-  ( (
ph  /\  w  e.  x )  ->  U. B  C_ recs
( F ) )
228eleq2d 2259 . . . . . 6  |-  ( ph  ->  ( w  e.  dom  U. B  <->  w  e.  x
) )
2322biimpar 297 . . . . 5  |-  ( (
ph  /\  w  e.  x )  ->  w  e.  dom  U. B )
24 funssfv 5557 . . . . 5  |-  ( ( Fun recs ( F )  /\  U. B  C_ recs ( F )  /\  w  e.  dom  U. B )  ->  (recs ( F ) `  w )  =  ( U. B `  w ) )
2520, 21, 23, 24syl3anc 1249 . . . 4  |-  ( (
ph  /\  w  e.  x )  ->  (recs ( F ) `  w
)  =  ( U. B `  w )
)
26 eloni 4390 . . . . . . . . 9  |-  ( x  e.  On  ->  Ord  x )
274, 26syl 14 . . . . . . . 8  |-  ( ph  ->  Ord  x )
28 ordelss 4394 . . . . . . . 8  |-  ( ( Ord  x  /\  w  e.  x )  ->  w  C_  x )
2927, 28sylan 283 . . . . . . 7  |-  ( (
ph  /\  w  e.  x )  ->  w  C_  x )
308adantr 276 . . . . . . 7  |-  ( (
ph  /\  w  e.  x )  ->  dom  U. B  =  x )
3129, 30sseqtrrd 3209 . . . . . 6  |-  ( (
ph  /\  w  e.  x )  ->  w  C_ 
dom  U. B )
32 fun2ssres 5275 . . . . . 6  |-  ( ( Fun recs ( F )  /\  U. B  C_ recs ( F )  /\  w  C_ 
dom  U. B )  -> 
(recs ( F )  |`  w )  =  ( U. B  |`  w
) )
3320, 21, 31, 32syl3anc 1249 . . . . 5  |-  ( (
ph  /\  w  e.  x )  ->  (recs ( F )  |`  w
)  =  ( U. B  |`  w ) )
3433fveq2d 5535 . . . 4  |-  ( (
ph  /\  w  e.  x )  ->  ( F `  (recs ( F )  |`  w
) )  =  ( F `  ( U. B  |`  w ) ) )
3518, 25, 343eqtr3d 2230 . . 3  |-  ( (
ph  /\  w  e.  x )  ->  ( U. B `  w )  =  ( F `  ( U. B  |`  w
) ) )
3635ralrimiva 2563 . 2  |-  ( ph  ->  A. w  e.  x  ( U. B `  w
)  =  ( F `
 ( U. B  |`  w ) ) )
37 fveq2 5531 . . . 4  |-  ( u  =  w  ->  ( U. B `  u )  =  ( U. B `  w ) )
38 reseq2 4917 . . . . 5  |-  ( u  =  w  ->  ( U. B  |`  u )  =  ( U. B  |`  w ) )
3938fveq2d 5535 . . . 4  |-  ( u  =  w  ->  ( F `  ( U. B  |`  u ) )  =  ( F `  ( U. B  |`  w
) ) )
4037, 39eqeq12d 2204 . . 3  |-  ( u  =  w  ->  (
( U. B `  u )  =  ( F `  ( U. B  |`  u ) )  <-> 
( U. B `  w )  =  ( F `  ( U. B  |`  w ) ) ) )
4140cbvralv 2718 . 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 134 1  |-  ( ph  ->  A. u  e.  x  ( U. B `  u
)  =  ( F `
 ( U. B  |`  u ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980   A.wal 1362    = wceq 1364   E.wex 1503    e. wcel 2160   {cab 2175   A.wral 2468   E.wrex 2469   _Vcvv 2752    u. cun 3142    C_ wss 3144   {csn 3607   <.cop 3610   U.cuni 3824   Ord word 4377   Oncon0 4378   dom cdm 4641    |` cres 4643   Fun wfun 5226    Fn wfn 5227   ` cfv 5232  recscrecs 6324
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-fv 5240  df-recs 6325
This theorem is referenced by:  tfrlemiex  6351
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