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Theorem tfrexlem 5858
Description: The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.)
Hypotheses
Ref Expression
tfrexlem.1  {  |  On  Fn  `  F `  |`  }
tfrexlem.2  Fun 
F  F `  _V
Assertion
Ref Expression
tfrexlem  C  V recs F `  C  _V
Distinct variable groups:   ,,,   , F,,
Allowed substitution hints:   (,,)    C(,,)    V(,,)

Proof of Theorem tfrexlem
Dummy variables  e  h  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5091 . . . . 5  C recs F ` recs F `  C
21eleq1d 2079 . . . 4  C recs F `  _V recs F `  C  _V
32imbi2d 219 . . 3  C recs F `  _V recs F `  C  _V
4 inss2 3126 . . . . . . 7  suc 
suc  i^i  On  C_  On
5 ssorduni 4151 . . . . . . 7  suc  suc  i^i  On  C_  On  Ord  U. suc  suc  i^i  On
64, 5ax-mp 7 . . . . . 6  Ord  U. suc  suc  i^i  On
7 vex 2529 . . . . . . . . . 10 
_V
87sucex 4163 . . . . . . . . 9  suc  _V
98sucex 4163 . . . . . . . 8  suc  suc  _V
109inex1 3854 . . . . . . 7  suc 
suc  i^i  On  _V
1110uniex 4112 . . . . . 6  U. suc  suc  i^i  On  _V
12 elon2 4051 . . . . . 6  U. suc  suc  i^i  On  On  Ord  U. suc  suc  i^i  On  U. suc  suc  i^i  On  _V
136, 11, 12mpbir2an 831 . . . . 5  U. suc  suc  i^i  On  On
14 tfrexlem.1 . . . . . . 7  {  |  On  Fn  `  F `  |`  }
1514tfrlem3 5836 . . . . . 6  {  |  On  Fn  `  F `  |`  }
16 tfrexlem.2 . . . . . . 7  Fun 
F  F `  _V
17 fveq2 5091 . . . . . . . . . 10  F `  F `
1817eleq1d 2079 . . . . . . . . 9  F `  _V  F `

_V
1918anbi2d 437 . . . . . . . 8  Fun  F  F `  _V  Fun 
F  F `  _V
2019cbvalv 1767 . . . . . . 7  Fun  F  F `  _V  Fun  F  F `  _V
2116, 20sylib 127 . . . . . 6  Fun 
F  F `  _V
2215, 21tfrlemi1 5855 . . . . 5  U. suc 
suc  i^i  On  On  Fn  U. suc 
suc  i^i  On 
U. suc  suc  i^i  On `  F `  |`
2313, 22mpan2 401 . . . 4  Fn  U. suc 
suc  i^i  On 
U. suc  suc  i^i  On `  F `  |`
2415recsfval 5841 . . . . . . . . . . 11 recs F  U.
2524breqi 3733 . . . . . . . . . 10 recs F  U.
26 df-br 3728 . . . . . . . . . 10 
U.  <. ,  >.  U.
27 eluni 3546 . . . . . . . . . 10  <. ,  >.  U.  h <. ,  >.  h  h
2825, 26, 273bitri 195 . . . . . . . . 9 recs F  h <. ,  >.  h  h
297sucid 4092 . . . . . . . . . . . . . . . . 17 
suc
30 simpr 103 . . . . . . . . . . . . . . . . . . . . . . . . . 26 
<. ,  >.  h  h  h
31 vex 2529 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  h 
_V
3214, 31tfrlem3a 5835 . . . . . . . . . . . . . . . . . . . . . . . . . 26  h  t  On  h  Fn  t  e  t  h `  e  F `  h  |`  e
3330, 32sylib 127 . . . . . . . . . . . . . . . . . . . . . . . . 25 
<. ,  >.  h  h  t  On  h  Fn  t  e  t  h `  e  F `  h  |`  e
34 simprl 468 . . . . . . . . . . . . . . . . . . . . . . . . . 26  <. , 
>.  h  h 
t  On  h  Fn  t  e  t  h `  e  F `  h  |`  e  t  On
35 simprrl 476 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  <. , 
>.  h  h 
t  On  h  Fn  t  e  t  h `  e  F `  h  |`  e  h  Fn  t
36 simpll 466 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  <. , 
>.  h  h 
t  On  h  Fn  t  e  t  h `  e  F `  h  |`  e  <. , 
>.  h
37 fnop 4916 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  h  Fn  t  <. ,  >.  h  t
3835, 36, 37syl2anc 391 . . . . . . . . . . . . . . . . . . . . . . . . . 26  <. , 
>.  h  h 
t  On  h  Fn  t  e  t  h `  e  F `  h  |`  e  t
39 onelon 4059 . . . . . . . . . . . . . . . . . . . . . . . . . 26  t  On  t  On
4034, 38, 39syl2anc 391 . . . . . . . . . . . . . . . . . . . . . . . . 25  <. , 
>.  h  h 
t  On  h  Fn  t  e  t  h `  e  F `  h  |`  e  On
4133, 40rexlimddv 2406 . . . . . . . . . . . . . . . . . . . . . . . 24 
<. ,  >.  h  h  On
4241adantl 262 . . . . . . . . . . . . . . . . . . . . . . 23  Fn  U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h  On
43 suceloni 4165 . . . . . . . . . . . . . . . . . . . . . . 23  On  suc  On
4442, 43syl 14 . . . . . . . . . . . . . . . . . . . . . 22  Fn  U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h 
suc  On
45 suceloni 4165 . . . . . . . . . . . . . . . . . . . . . 22  suc  On  suc 
suc  On
4644, 45syl 14 . . . . . . . . . . . . . . . . . . . . 21  Fn  U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h 
suc  suc  On
47 onss 4157 . . . . . . . . . . . . . . . . . . . . 21  suc 
suc  On  suc  suc  C_  On
4846, 47syl 14 . . . . . . . . . . . . . . . . . . . 20  Fn  U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h 
suc  suc  C_  On
49 df-ss 2899 . . . . . . . . . . . . . . . . . . . 20  suc 
suc  C_  On  suc  suc  i^i  On  suc  suc
5048, 49sylib 127 . . . . . . . . . . . . . . . . . . 19  Fn  U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h  suc  suc  i^i  On  suc  suc
5150unieqd 3554 . . . . . . . . . . . . . . . . . 18  Fn  U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h  U. suc  suc  i^i  On  U. suc  suc
52 eloni 4050 . . . . . . . . . . . . . . . . . . . 20  suc  On  Ord 
suc
53 ordtr 4053 . . . . . . . . . . . . . . . . . . . 20  Ord 
suc  Tr  suc
5444, 52, 533syl 17 . . . . . . . . . . . . . . . . . . 19  Fn  U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h 
Tr  suc
558unisuc 4088 . . . . . . . . . . . . . . . . . . 19  Tr 
suc  U. suc  suc  suc
5654, 55sylib 127 . . . . . . . . . . . . . . . . . 18  Fn  U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h  U. suc  suc  suc
5751, 56eqtrd 2045 . . . . . . . . . . . . . . . . 17  Fn  U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h  U. suc  suc  i^i  On  suc
5829, 57syl5eleqr 2100 . . . . . . . . . . . . . . . 16  Fn  U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h  U. suc  suc  i^i  On
59 fndm 4912 . . . . . . . . . . . . . . . . 17  Fn  U. suc 
suc  i^i  On  dom 
U. suc  suc  i^i  On
6059ad2antrr 457 . . . . . . . . . . . . . . . 16  Fn  U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h 
dom  U. suc  suc  i^i  On
6158, 60eleqtrrd 2090 . . . . . . . . . . . . . . 15  Fn  U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h  dom
627eldm 4447 . . . . . . . . . . . . . . 15  dom
6361, 62sylib 127 . . . . . . . . . . . . . 14  Fn  U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h
64 simpr 103 . . . . . . . . . . . . . . 15  Fn 
U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h
65 fneq2 4902 . . . . . . . . . . . . . . . . . . . . 21  U. suc 
suc  i^i  On  Fn  Fn  U. suc  suc  i^i  On
66 raleq 2474 . . . . . . . . . . . . . . . . . . . . 21  U. suc 
suc  i^i  On  `  F `  |`  U. suc  suc  i^i  On `  F `  |`
6765, 66anbi12d 442 . . . . . . . . . . . . . . . . . . . 20  U. suc 
suc  i^i  On  Fn  `  F `  |`  Fn 
U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`
6867rspcev 2624 . . . . . . . . . . . . . . . . . . 19 
U. suc  suc  i^i  On  On  Fn  U. suc 
suc  i^i  On 
U. suc  suc  i^i  On `  F `  |`  On  Fn  `  F `  |`
6913, 68mpan 400 . . . . . . . . . . . . . . . . . 18  Fn  U. suc  suc  i^i  On 
U. suc  suc  i^i  On `  F `  |`  On  Fn  `  F `  |`
70 vex 2529 . . . . . . . . . . . . . . . . . . 19 
_V
7114, 70tfrlem3a 5835 . . . . . . . . . . . . . . . . . 18  On  Fn  `  F `  |`
7269, 71sylibr 137 . . . . . . . . . . . . . . . . 17  Fn  U. suc  suc  i^i  On 
U. suc  suc  i^i  On `  F `  |`
7372ad2antrr 457 . . . . . . . . . . . . . . . 16  Fn 
U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h
74 simplrr 473 . . . . . . . . . . . . . . . 16  Fn 
U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h 
h
75 simplrl 472 . . . . . . . . . . . . . . . . 17  Fn 
U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h  <. ,  >.  h
76 df-br 3728 . . . . . . . . . . . . . . . . 17  h  <. ,  >.  h
7775, 76sylibr 137 . . . . . . . . . . . . . . . 16  Fn 
U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h  h
7815tfrlem5 5840 . . . . . . . . . . . . . . . . 17  h  h
7978imp 115 . . . . . . . . . . . . . . . 16  h  h
8073, 74, 64, 77, 79syl22anc 1117 . . . . . . . . . . . . . . 15  Fn 
U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h
8164, 80breqtrd 3751 . . . . . . . . . . . . . 14  Fn 
U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h
8263, 81exlimddv 1751 . . . . . . . . . . . . 13  Fn  U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h
83 vex 2529 . . . . . . . . . . . . . 14 
_V
847, 83brelrn 4482 . . . . . . . . . . . . 13  ran
8582, 84syl 14 . . . . . . . . . . . 12  Fn  U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h  ran
86 elssuni 3571 . . . . . . . . . . . 12  ran  C_  U. ran
8785, 86syl 14 . . . . . . . . . . 11  Fn  U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |`  <. ,  >.  h  h  C_  U. ran
8887ex 108 . . . . . . . . . 10  Fn  U. suc  suc  i^i  On 
U. suc  suc  i^i  On `  F `  |`  <. , 
>.  h  h  C_ 
U. ran
8988exlimdv 1673 . . . . . . . . 9  Fn  U. suc  suc  i^i  On 
U. suc  suc  i^i  On `  F `  |`  h <. ,  >.  h  h  C_  U. ran
9028, 89syl5bi 141 . . . . . . . 8  Fn  U. suc  suc  i^i  On 
U. suc  suc  i^i  On `  F `  |` recs F  C_  U. ran
9190alrimiv 1727 . . . . . . 7  Fn  U. suc  suc  i^i  On 
U. suc  suc  i^i  On `  F `  |` recs F  C_ 
U. ran
92 fvss 5102 . . . . . . 7 recs F  C_ 
U. ran recs F `
 C_  U.
ran
9391, 92syl 14 . . . . . 6  Fn  U. suc  suc  i^i  On 
U. suc  suc  i^i  On `  F `  |` recs F `  C_  U. ran
9470rnex 4514 . . . . . . . 8  ran  _V
9594uniex 4112 . . . . . . 7  U. ran  _V
9695ssex 3857 . . . . . 6 recs F `  C_  U. ran recs F `  _V
9793, 96syl 14 . . . . 5  Fn  U. suc  suc  i^i  On 
U. suc  suc  i^i  On `  F `  |` recs F `  _V
9897exlimiv 1462 . . . 4  Fn  U. suc  suc  i^i  On  U. suc  suc  i^i  On `  F `  |` recs F `  _V
9923, 98syl 14 . . 3 recs F `

_V
1003, 99vtoclg 2581 . 2  C  V recs F `  C  _V
101100impcom 116 1  C  V recs F `  C  _V
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97  wal 1221   wceq 1223  wex 1354   wcel 1366   {cab 1999  wral 2275  wrex 2276   _Vcvv 2526    i^i cin 2884    C_ wss 2885   <.cop 3342   U.cuni 3543   class class class wbr 3727   Tr wtr 3817   Ord word 4037   Oncon0 4038   suc csuc 4040   dom cdm 4260   ran crn 4261    |` cres 4262   Fun wfun 4811    Fn wfn 4812   ` cfv 4817  recscrecs 5829
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 529  ax-in2 530  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-13 1377  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-coll 3835  ax-sep 3838  ax-pow 3890  ax-pr 3907  ax-un 4108  ax-setind 4192
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-fal 1229  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ne 2179  df-ral 2280  df-rex 2281  df-reu 2282  df-rab 2284  df-v 2528  df-sbc 2733  df-csb 2821  df-dif 2888  df-un 2890  df-in 2892  df-ss 2899  df-nul 3193  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-iun 3622  df-br 3728  df-opab 3782  df-mpt 3783  df-tr 3818  df-id 3993  df-iord 4041  df-on 4043  df-suc 4046  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-iota 4782  df-fun 4819  df-fn 4820  df-f 4821  df-f1 4822  df-fo 4823  df-f1o 4824  df-fv 4825  df-recs 5830
This theorem is referenced by:  tfrex  5864
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