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Theorem tfrlemiubacc 6334
Description: The union of  B satisfies the recursion rule (lemma for tfrlemi1 6336). (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 6332 . . . . . . . 8  |-  ( ph  ->  U. B  Fn  x
)
7 fndm 5317 . . . . . . . 8  |-  ( U. B  Fn  x  ->  dom  U. B  =  x
)
86, 7syl 14 . . . . . . 7  |-  ( ph  ->  dom  U. B  =  x )
91, 2, 3, 4, 5tfrlemibacc 6330 . . . . . . . . . 10  |-  ( ph  ->  B  C_  A )
109unissd 3835 . . . . . . . . 9  |-  ( ph  ->  U. B  C_  U. A
)
111recsfval 6319 . . . . . . . . 9  |- recs ( F )  =  U. A
1210, 11sseqtrrdi 3206 . . . . . . . 8  |-  ( ph  ->  U. B  C_ recs ( F ) )
13 dmss 4828 . . . . . . . 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 3194 . . . . . 6  |-  ( ph  ->  x  C_  dom recs ( F ) )
1615sselda 3157 . . . . 5  |-  ( (
ph  /\  w  e.  x )  ->  w  e.  dom recs ( F ) )
171tfrlem9 6323 . . . . 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 6321 . . . . . 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 2247 . . . . . 6  |-  ( ph  ->  ( w  e.  dom  U. B  <->  w  e.  x
) )
2322biimpar 297 . . . . 5  |-  ( (
ph  /\  w  e.  x )  ->  w  e.  dom  U. B )
24 funssfv 5543 . . . . 5  |-  ( ( Fun recs ( F )  /\  U. B  C_ recs ( F )  /\  w  e.  dom  U. B )  ->  (recs ( F ) `  w )  =  ( U. B `  w ) )
2520, 21, 23, 24syl3anc 1238 . . . 4  |-  ( (
ph  /\  w  e.  x )  ->  (recs ( F ) `  w
)  =  ( U. B `  w )
)
26 eloni 4377 . . . . . . . . 9  |-  ( x  e.  On  ->  Ord  x )
274, 26syl 14 . . . . . . . 8  |-  ( ph  ->  Ord  x )
28 ordelss 4381 . . . . . . . 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 3196 . . . . . 6  |-  ( (
ph  /\  w  e.  x )  ->  w  C_ 
dom  U. B )
32 fun2ssres 5261 . . . . . 6  |-  ( ( Fun recs ( F )  /\  U. B  C_ recs ( F )  /\  w  C_ 
dom  U. B )  -> 
(recs ( F )  |`  w )  =  ( U. B  |`  w
) )
3320, 21, 31, 32syl3anc 1238 . . . . 5  |-  ( (
ph  /\  w  e.  x )  ->  (recs ( F )  |`  w
)  =  ( U. B  |`  w ) )
3433fveq2d 5521 . . . 4  |-  ( (
ph  /\  w  e.  x )  ->  ( F `  (recs ( F )  |`  w
) )  =  ( F `  ( U. B  |`  w ) ) )
3518, 25, 343eqtr3d 2218 . . 3  |-  ( (
ph  /\  w  e.  x )  ->  ( U. B `  w )  =  ( F `  ( U. B  |`  w
) ) )
3635ralrimiva 2550 . 2  |-  ( ph  ->  A. w  e.  x  ( U. B `  w
)  =  ( F `
 ( U. B  |`  w ) ) )
37 fveq2 5517 . . . 4  |-  ( u  =  w  ->  ( U. B `  u )  =  ( U. B `  w ) )
38 reseq2 4904 . . . . 5  |-  ( u  =  w  ->  ( U. B  |`  u )  =  ( U. B  |`  w ) )
3938fveq2d 5521 . . . 4  |-  ( u  =  w  ->  ( F `  ( U. B  |`  u ) )  =  ( F `  ( U. B  |`  w
) ) )
4037, 39eqeq12d 2192 . . 3  |-  ( u  =  w  ->  (
( U. B `  u )  =  ( F `  ( U. B  |`  u ) )  <-> 
( U. B `  w )  =  ( F `  ( U. B  |`  w ) ) ) )
4140cbvralv 2705 . 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 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    C_ wss 3131   {csn 3594   <.cop 3597   U.cuni 3811   Ord word 4364   Oncon0 4365   dom cdm 4628    |` cres 4630   Fun wfun 5212    Fn wfn 5213   ` cfv 5218  recscrecs 6308
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-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-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-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-recs 6309
This theorem is referenced by:  tfrlemiex  6335
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