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Theorem tfrlemiubacc 5944
Description: The union of  B satisfies the recursion rule (lemma for tfrlemi1 5946). (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 5942 . . . . . . . 8  |-  ( ph  ->  U. B  Fn  x
)
7 fndm 4998 . . . . . . . 8  |-  ( U. B  Fn  x  ->  dom  U. B  =  x
)
86, 7syl 14 . . . . . . 7  |-  ( ph  ->  dom  U. B  =  x )
91, 2, 3, 4, 5tfrlemibacc 5940 . . . . . . . . . 10  |-  ( ph  ->  B  C_  A )
109unissd 3604 . . . . . . . . 9  |-  ( ph  ->  U. B  C_  U. A
)
111recsfval 5931 . . . . . . . . 9  |- recs ( F )  =  U. A
1210, 11syl6sseqr 2992 . . . . . . . 8  |-  ( ph  ->  U. B  C_ recs ( F ) )
13 dmss 4534 . . . . . . . 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 2980 . . . . . 6  |-  ( ph  ->  x  C_  dom recs ( F ) )
1615sselda 2945 . . . . 5  |-  ( (
ph  /\  w  e.  x )  ->  w  e.  dom recs ( F ) )
171tfrlem9 5935 . . . . 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 5933 . . . . . 6  |-  Fun recs ( F )
2019a1i 9 . . . . 5  |-  ( (
ph  /\  w  e.  x )  ->  Fun recs ( F ) )
2112adantr 261 . . . . 5  |-  ( (
ph  /\  w  e.  x )  ->  U. B  C_ recs
( F ) )
228eleq2d 2107 . . . . . 6  |-  ( ph  ->  ( w  e.  dom  U. B  <->  w  e.  x
) )
2322biimpar 281 . . . . 5  |-  ( (
ph  /\  w  e.  x )  ->  w  e.  dom  U. B )
24 funssfv 5199 . . . . 5  |-  ( ( Fun recs ( F )  /\  U. B  C_ recs ( F )  /\  w  e.  dom  U. B )  ->  (recs ( F ) `  w )  =  ( U. B `  w ) )
2520, 21, 23, 24syl3anc 1135 . . . 4  |-  ( (
ph  /\  w  e.  x )  ->  (recs ( F ) `  w
)  =  ( U. B `  w )
)
26 eloni 4112 . . . . . . . . 9  |-  ( x  e.  On  ->  Ord  x )
274, 26syl 14 . . . . . . . 8  |-  ( ph  ->  Ord  x )
28 ordelss 4116 . . . . . . . 8  |-  ( ( Ord  x  /\  w  e.  x )  ->  w  C_  x )
2927, 28sylan 267 . . . . . . 7  |-  ( (
ph  /\  w  e.  x )  ->  w  C_  x )
308adantr 261 . . . . . . 7  |-  ( (
ph  /\  w  e.  x )  ->  dom  U. B  =  x )
3129, 30sseqtr4d 2982 . . . . . 6  |-  ( (
ph  /\  w  e.  x )  ->  w  C_ 
dom  U. B )
32 fun2ssres 4943 . . . . . 6  |-  ( ( Fun recs ( F )  /\  U. B  C_ recs ( F )  /\  w  C_ 
dom  U. B )  -> 
(recs ( F )  |`  w )  =  ( U. B  |`  w
) )
3320, 21, 31, 32syl3anc 1135 . . . . 5  |-  ( (
ph  /\  w  e.  x )  ->  (recs ( F )  |`  w
)  =  ( U. B  |`  w ) )
3433fveq2d 5182 . . . 4  |-  ( (
ph  /\  w  e.  x )  ->  ( F `  (recs ( F )  |`  w
) )  =  ( F `  ( U. B  |`  w ) ) )
3518, 25, 343eqtr3d 2080 . . 3  |-  ( (
ph  /\  w  e.  x )  ->  ( U. B `  w )  =  ( F `  ( U. B  |`  w
) ) )
3635ralrimiva 2392 . 2  |-  ( ph  ->  A. w  e.  x  ( U. B `  w
)  =  ( F `
 ( U. B  |`  w ) ) )
37 fveq2 5178 . . . 4  |-  ( u  =  w  ->  ( U. B `  u )  =  ( U. B `  w ) )
38 reseq2 4607 . . . . 5  |-  ( u  =  w  ->  ( U. B  |`  u )  =  ( U. B  |`  w ) )
3938fveq2d 5182 . . . 4  |-  ( u  =  w  ->  ( F `  ( U. B  |`  u ) )  =  ( F `  ( U. B  |`  w
) ) )
4037, 39eqeq12d 2054 . . 3  |-  ( u  =  w  ->  (
( U. B `  u )  =  ( F `  ( U. B  |`  u ) )  <-> 
( U. B `  w )  =  ( F `  ( U. B  |`  w ) ) ) )
4140cbvralv 2533 . 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 137 1  |-  ( ph  ->  A. u  e.  x  ( U. B `  u
)  =  ( F `
 ( U. B  |`  u ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    /\ w3a 885   A.wal 1241    = wceq 1243   E.wex 1381    e. wcel 1393   {cab 2026   A.wral 2306   E.wrex 2307   _Vcvv 2557    u. cun 2915    C_ wss 2917   {csn 3375   <.cop 3378   U.cuni 3580   Ord word 4099   Oncon0 4100   dom cdm 4345    |` cres 4347   Fun wfun 4896    Fn wfn 4897   ` cfv 4902  recscrecs 5919
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-recs 5920
This theorem is referenced by:  tfrlemiex  5945
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