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Theorem aomclem4 30931
Description: Lemma for dfac11 30936. Limit case. Patch together well-orderings constructed so far using fnwe2 30927 to cover the limit rank. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Hypotheses
Ref Expression
aomclem4.f  |-  F  =  { <. a ,  b
>.  |  ( ( rank `  a )  _E  ( rank `  b
)  \/  ( (
rank `  a )  =  ( rank `  b
)  /\  a (
z `  suc  ( rank `  a ) ) b ) ) }
aomclem4.on  |-  ( ph  ->  dom  z  e.  On )
aomclem4.su  |-  ( ph  ->  dom  z  =  U. dom  z )
aomclem4.we  |-  ( ph  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )
Assertion
Ref Expression
aomclem4  |-  ( ph  ->  F  We  ( R1
`  dom  z )
)
Distinct variable groups:    z, a,
b    ph, a, b
Allowed substitution hints:    ph( z)    F( z, a, b)

Proof of Theorem aomclem4
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 suceq 4949 . . 3  |-  ( c  =  ( rank `  a
)  ->  suc  c  =  suc  ( rank `  a
) )
21fveq2d 5876 . 2  |-  ( c  =  ( rank `  a
)  ->  ( z `  suc  c )  =  ( z `  suc  ( rank `  a )
) )
3 aomclem4.f . 2  |-  F  =  { <. a ,  b
>.  |  ( ( rank `  a )  _E  ( rank `  b
)  \/  ( (
rank `  a )  =  ( rank `  b
)  /\  a (
z `  suc  ( rank `  a ) ) b ) ) }
4 r1fnon 8197 . . . . . . . . . . . . . 14  |-  R1  Fn  On
5 fnfun 5684 . . . . . . . . . . . . . 14  |-  ( R1  Fn  On  ->  Fun  R1 )
64, 5ax-mp 5 . . . . . . . . . . . . 13  |-  Fun  R1
7 fndm 5686 . . . . . . . . . . . . . . 15  |-  ( R1  Fn  On  ->  dom  R1  =  On )
84, 7ax-mp 5 . . . . . . . . . . . . . 14  |-  dom  R1  =  On
98eqimss2i 3564 . . . . . . . . . . . . 13  |-  On  C_  dom  R1
106, 9pm3.2i 455 . . . . . . . . . . . 12  |-  ( Fun 
R1  /\  On  C_  dom  R1 )
11 aomclem4.on . . . . . . . . . . . 12  |-  ( ph  ->  dom  z  e.  On )
12 funfvima2 6147 . . . . . . . . . . . 12  |-  ( ( Fun  R1  /\  On  C_ 
dom  R1 )  ->  ( dom  z  e.  On  ->  ( R1 `  dom  z )  e.  ( R1 " On ) ) )
1310, 11, 12mpsyl 63 . . . . . . . . . . 11  |-  ( ph  ->  ( R1 `  dom  z )  e.  ( R1 " On ) )
14 elssuni 4281 . . . . . . . . . . 11  |-  ( ( R1 `  dom  z
)  e.  ( R1
" On )  -> 
( R1 `  dom  z )  C_  U. ( R1 " On ) )
1513, 14syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( R1 `  dom  z )  C_  U. ( R1 " On ) )
1615sselda 3509 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  ( R1 `  dom  z
) )  ->  b  e.  U. ( R1 " On ) )
17 rankidb 8230 . . . . . . . . 9  |-  ( b  e.  U. ( R1
" On )  -> 
b  e.  ( R1
`  suc  ( rank `  b ) ) )
1816, 17syl 16 . . . . . . . 8  |-  ( (
ph  /\  b  e.  ( R1 `  dom  z
) )  ->  b  e.  ( R1 `  suc  ( rank `  b )
) )
19 suceq 4949 . . . . . . . . . 10  |-  ( (
rank `  b )  =  ( rank `  a
)  ->  suc  ( rank `  b )  =  suc  ( rank `  a )
)
2019fveq2d 5876 . . . . . . . . 9  |-  ( (
rank `  b )  =  ( rank `  a
)  ->  ( R1 ` 
suc  ( rank `  b
) )  =  ( R1 `  suc  ( rank `  a ) ) )
2120eleq2d 2537 . . . . . . . 8  |-  ( (
rank `  b )  =  ( rank `  a
)  ->  ( b  e.  ( R1 `  suc  ( rank `  b )
)  <->  b  e.  ( R1 `  suc  ( rank `  a ) ) ) )
2218, 21syl5ibcom 220 . . . . . . 7  |-  ( (
ph  /\  b  e.  ( R1 `  dom  z
) )  ->  (
( rank `  b )  =  ( rank `  a
)  ->  b  e.  ( R1 `  suc  ( rank `  a ) ) ) )
2322expimpd 603 . . . . . 6  |-  ( ph  ->  ( ( b  e.  ( R1 `  dom  z )  /\  ( rank `  b )  =  ( rank `  a
) )  ->  b  e.  ( R1 `  suc  ( rank `  a )
) ) )
2423ss2abdv 3578 . . . . 5  |-  ( ph  ->  { b  |  ( b  e.  ( R1
`  dom  z )  /\  ( rank `  b
)  =  ( rank `  a ) ) } 
C_  { b  |  b  e.  ( R1
`  suc  ( rank `  a ) ) } )
25 df-rab 2826 . . . . 5  |-  { b  e.  ( R1 `  dom  z )  |  (
rank `  b )  =  ( rank `  a
) }  =  {
b  |  ( b  e.  ( R1 `  dom  z )  /\  ( rank `  b )  =  ( rank `  a
) ) }
26 abid2 2607 . . . . . 6  |-  { b  |  b  e.  ( R1 `  suc  ( rank `  a ) ) }  =  ( R1
`  suc  ( rank `  a ) )
2726eqcomi 2480 . . . . 5  |-  ( R1
`  suc  ( rank `  a ) )  =  { b  |  b  e.  ( R1 `  suc  ( rank `  a
) ) }
2824, 25, 273sstr4g 3550 . . . 4  |-  ( ph  ->  { b  e.  ( R1 `  dom  z
)  |  ( rank `  b )  =  (
rank `  a ) }  C_  ( R1 `  suc  ( rank `  a
) ) )
2928adantr 465 . . 3  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  { b  e.  ( R1 `  dom  z )  |  (
rank `  b )  =  ( rank `  a
) }  C_  ( R1 `  suc  ( rank `  a ) ) )
30 rankr1ai 8228 . . . . . 6  |-  ( a  e.  ( R1 `  dom  z )  ->  ( rank `  a )  e. 
dom  z )
3130adantl 466 . . . . 5  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  ( rank `  a )  e. 
dom  z )
32 eloni 4894 . . . . . . . 8  |-  ( dom  z  e.  On  ->  Ord 
dom  z )
3311, 32syl 16 . . . . . . 7  |-  ( ph  ->  Ord  dom  z )
34 aomclem4.su . . . . . . 7  |-  ( ph  ->  dom  z  =  U. dom  z )
35 limsuc2 30914 . . . . . . 7  |-  ( ( Ord  dom  z  /\  dom  z  =  U. dom  z )  ->  (
( rank `  a )  e.  dom  z  <->  suc  ( rank `  a )  e.  dom  z ) )
3633, 34, 35syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( rank `  a
)  e.  dom  z  <->  suc  ( rank `  a
)  e.  dom  z
) )
3736adantr 465 . . . . 5  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  (
( rank `  a )  e.  dom  z  <->  suc  ( rank `  a )  e.  dom  z ) )
3831, 37mpbid 210 . . . 4  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  suc  ( rank `  a )  e.  dom  z )
39 aomclem4.we . . . . . 6  |-  ( ph  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )
40 fveq2 5872 . . . . . . . 8  |-  ( a  =  b  ->  (
z `  a )  =  ( z `  b ) )
41 fveq2 5872 . . . . . . . 8  |-  ( a  =  b  ->  ( R1 `  a )  =  ( R1 `  b
) )
4240, 41weeq12d 30913 . . . . . . 7  |-  ( a  =  b  ->  (
( z `  a
)  We  ( R1
`  a )  <->  ( z `  b )  We  ( R1 `  b ) ) )
4342cbvralv 3093 . . . . . 6  |-  ( A. a  e.  dom  z ( z `  a )  We  ( R1 `  a )  <->  A. b  e.  dom  z ( z `
 b )  We  ( R1 `  b
) )
4439, 43sylib 196 . . . . 5  |-  ( ph  ->  A. b  e.  dom  z ( z `  b )  We  ( R1 `  b ) )
4544adantr 465 . . . 4  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  A. b  e.  dom  z ( z `
 b )  We  ( R1 `  b
) )
46 fveq2 5872 . . . . . 6  |-  ( b  =  suc  ( rank `  a )  ->  (
z `  b )  =  ( z `  suc  ( rank `  a
) ) )
47 fveq2 5872 . . . . . 6  |-  ( b  =  suc  ( rank `  a )  ->  ( R1 `  b )  =  ( R1 `  suc  ( rank `  a )
) )
4846, 47weeq12d 30913 . . . . 5  |-  ( b  =  suc  ( rank `  a )  ->  (
( z `  b
)  We  ( R1
`  b )  <->  ( z `  suc  ( rank `  a
) )  We  ( R1 `  suc  ( rank `  a ) ) ) )
4948rspcva 3217 . . . 4  |-  ( ( suc  ( rank `  a
)  e.  dom  z  /\  A. b  e.  dom  z ( z `  b )  We  ( R1 `  b ) )  ->  ( z `  suc  ( rank `  a
) )  We  ( R1 `  suc  ( rank `  a ) ) )
5038, 45, 49syl2anc 661 . . 3  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  (
z `  suc  ( rank `  a ) )  We  ( R1 `  suc  ( rank `  a )
) )
51 wess 4872 . . 3  |-  ( { b  e.  ( R1
`  dom  z )  |  ( rank `  b
)  =  ( rank `  a ) }  C_  ( R1 `  suc  ( rank `  a ) )  ->  ( ( z `
 suc  ( rank `  a ) )  We  ( R1 `  suc  ( rank `  a )
)  ->  ( z `  suc  ( rank `  a
) )  We  {
b  e.  ( R1
`  dom  z )  |  ( rank `  b
)  =  ( rank `  a ) } ) )
5229, 50, 51sylc 60 . 2  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  (
z `  suc  ( rank `  a ) )  We 
{ b  e.  ( R1 `  dom  z
)  |  ( rank `  b )  =  (
rank `  a ) } )
53 rankf 8224 . . . 4  |-  rank : U. ( R1 " On ) --> On
5453a1i 11 . . 3  |-  ( ph  ->  rank : U. ( R1 " On ) --> On )
55 fssres 5757 . . 3  |-  ( (
rank : U. ( R1
" On ) --> On 
/\  ( R1 `  dom  z )  C_  U. ( R1 " On ) )  ->  ( rank  |`  ( R1 `  dom  z ) ) : ( R1
`  dom  z ) --> On )
5654, 15, 55syl2anc 661 . 2  |-  ( ph  ->  ( rank  |`  ( R1
`  dom  z )
) : ( R1
`  dom  z ) --> On )
57 epweon 6614 . . 3  |-  _E  We  On
5857a1i 11 . 2  |-  ( ph  ->  _E  We  On )
592, 3, 52, 56, 58fnwe2 30927 1  |-  ( ph  ->  F  We  ( R1
`  dom  z )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2817   {crab 2821    C_ wss 3481   U.cuni 4251   class class class wbr 4453   {copab 4510    _E cep 4795    We wwe 4843   Ord word 4883   Oncon0 4884   suc csuc 4886   dom cdm 5005    |` cres 5007   "cima 5008   Fun wfun 5588    Fn wfn 5589   -->wf 5590   ` cfv 5594   R1cr1 8192   rankcrnk 8193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6696  df-recs 7054  df-rdg 7088  df-r1 8194  df-rank 8195
This theorem is referenced by:  aomclem5  30932
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