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Theorem aomclem4 27123
Description: Lemma for dfac11 27128. Limit case. Patch together well-orderings constructed so far using fnwe2 27119 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 4638 . . 3  |-  ( c  =  ( rank `  a
)  ->  suc  c  =  suc  ( rank `  a
) )
21fveq2d 5724 . 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 7685 . . . . . . . . . . . . . 14  |-  R1  Fn  On
5 fnfun 5534 . . . . . . . . . . . . . 14  |-  ( R1  Fn  On  ->  Fun  R1 )
64, 5ax-mp 8 . . . . . . . . . . . . 13  |-  Fun  R1
7 fndm 5536 . . . . . . . . . . . . . . 15  |-  ( R1  Fn  On  ->  dom  R1  =  On )
84, 7ax-mp 8 . . . . . . . . . . . . . 14  |-  dom  R1  =  On
98eqimss2i 3395 . . . . . . . . . . . . 13  |-  On  C_  dom  R1
106, 9pm3.2i 442 . . . . . . . . . . . 12  |-  ( Fun 
R1  /\  On  C_  dom  R1 )
11 aomclem4.on . . . . . . . . . . . 12  |-  ( ph  ->  dom  z  e.  On )
12 funfvima2 5966 . . . . . . . . . . . 12  |-  ( ( Fun  R1  /\  On  C_ 
dom  R1 )  ->  ( dom  z  e.  On  ->  ( R1 `  dom  z )  e.  ( R1 " On ) ) )
1310, 11, 12mpsyl 61 . . . . . . . . . . 11  |-  ( ph  ->  ( R1 `  dom  z )  e.  ( R1 " On ) )
14 elssuni 4035 . . . . . . . . . . 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 3340 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  ( R1 `  dom  z
) )  ->  b  e.  U. ( R1 " On ) )
17 rankidb 7718 . . . . . . . . 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 4638 . . . . . . . . . 10  |-  ( (
rank `  b )  =  ( rank `  a
)  ->  suc  ( rank `  b )  =  suc  ( rank `  a )
)
2019fveq2d 5724 . . . . . . . . 9  |-  ( (
rank `  b )  =  ( rank `  a
)  ->  ( R1 ` 
suc  ( rank `  b
) )  =  ( R1 `  suc  ( rank `  a ) ) )
2120eleq2d 2502 . . . . . . . 8  |-  ( (
rank `  b )  =  ( rank `  a
)  ->  ( b  e.  ( R1 `  suc  ( rank `  b )
)  <->  b  e.  ( R1 `  suc  ( rank `  a ) ) ) )
2218, 21syl5ibcom 212 . . . . . . 7  |-  ( (
ph  /\  b  e.  ( R1 `  dom  z
) )  ->  (
( rank `  b )  =  ( rank `  a
)  ->  b  e.  ( R1 `  suc  ( rank `  a ) ) ) )
2322expimpd 587 . . . . . 6  |-  ( ph  ->  ( ( b  e.  ( R1 `  dom  z )  /\  ( rank `  b )  =  ( rank `  a
) )  ->  b  e.  ( R1 `  suc  ( rank `  a )
) ) )
2423ss2abdv 3408 . . . . 5  |-  ( ph  ->  { b  |  ( b  e.  ( R1
`  dom  z )  /\  ( rank `  b
)  =  ( rank `  a ) ) } 
C_  { b  |  b  e.  ( R1
`  suc  ( rank `  a ) ) } )
25 df-rab 2706 . . . . 5  |-  { b  e.  ( R1 `  dom  z )  |  (
rank `  b )  =  ( rank `  a
) }  =  {
b  |  ( b  e.  ( R1 `  dom  z )  /\  ( rank `  b )  =  ( rank `  a
) ) }
26 abid2 2552 . . . . . 6  |-  { b  |  b  e.  ( R1 `  suc  ( rank `  a ) ) }  =  ( R1
`  suc  ( rank `  a ) )
2726eqcomi 2439 . . . . 5  |-  ( R1
`  suc  ( rank `  a ) )  =  { b  |  b  e.  ( R1 `  suc  ( rank `  a
) ) }
2824, 25, 273sstr4g 3381 . . . 4  |-  ( ph  ->  { b  e.  ( R1 `  dom  z
)  |  ( rank `  b )  =  (
rank `  a ) }  C_  ( R1 `  suc  ( rank `  a
) ) )
2928adantr 452 . . 3  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  { b  e.  ( R1 `  dom  z )  |  (
rank `  b )  =  ( rank `  a
) }  C_  ( R1 `  suc  ( rank `  a ) ) )
30 rankr1ai 7716 . . . . . 6  |-  ( a  e.  ( R1 `  dom  z )  ->  ( rank `  a )  e. 
dom  z )
3130adantl 453 . . . . 5  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  ( rank `  a )  e. 
dom  z )
32 eloni 4583 . . . . . . . 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 27106 . . . . . . 7  |-  ( ( Ord  dom  z  /\  dom  z  =  U. dom  z )  ->  (
( rank `  a )  e.  dom  z  <->  suc  ( rank `  a )  e.  dom  z ) )
3633, 34, 35syl2anc 643 . . . . . 6  |-  ( ph  ->  ( ( rank `  a
)  e.  dom  z  <->  suc  ( rank `  a
)  e.  dom  z
) )
3736adantr 452 . . . . 5  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  (
( rank `  a )  e.  dom  z  <->  suc  ( rank `  a )  e.  dom  z ) )
3831, 37mpbid 202 . . . 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 5720 . . . . . . . 8  |-  ( a  =  b  ->  (
z `  a )  =  ( z `  b ) )
41 fveq2 5720 . . . . . . . 8  |-  ( a  =  b  ->  ( R1 `  a )  =  ( R1 `  b
) )
4240, 41weeq12d 27105 . . . . . . 7  |-  ( a  =  b  ->  (
( z `  a
)  We  ( R1
`  a )  <->  ( z `  b )  We  ( R1 `  b ) ) )
4342cbvralv 2924 . . . . . 6  |-  ( A. a  e.  dom  z ( z `  a )  We  ( R1 `  a )  <->  A. b  e.  dom  z ( z `
 b )  We  ( R1 `  b
) )
4439, 43sylib 189 . . . . 5  |-  ( ph  ->  A. b  e.  dom  z ( z `  b )  We  ( R1 `  b ) )
4544adantr 452 . . . 4  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  A. b  e.  dom  z ( z `
 b )  We  ( R1 `  b
) )
46 fveq2 5720 . . . . . 6  |-  ( b  =  suc  ( rank `  a )  ->  (
z `  b )  =  ( z `  suc  ( rank `  a
) ) )
47 fveq2 5720 . . . . . 6  |-  ( b  =  suc  ( rank `  a )  ->  ( R1 `  b )  =  ( R1 `  suc  ( rank `  a )
) )
4846, 47weeq12d 27105 . . . . 5  |-  ( b  =  suc  ( rank `  a )  ->  (
( z `  b
)  We  ( R1
`  b )  <->  ( z `  suc  ( rank `  a
) )  We  ( R1 `  suc  ( rank `  a ) ) ) )
4948rspcva 3042 . . . 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 643 . . 3  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  (
z `  suc  ( rank `  a ) )  We  ( R1 `  suc  ( rank `  a )
) )
51 wess 4561 . . 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 58 . 2  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  (
z `  suc  ( rank `  a ) )  We 
{ b  e.  ( R1 `  dom  z
)  |  ( rank `  b )  =  (
rank `  a ) } )
53 rankf 7712 . . . 4  |-  rank : U. ( R1 " On ) --> On
5453a1i 11 . . 3  |-  ( ph  ->  rank : U. ( R1 " On ) --> On )
55 fssres 5602 . . 3  |-  ( (
rank : U. ( R1
" On ) --> On 
/\  ( R1 `  dom  z )  C_  U. ( R1 " On ) )  ->  ( rank  |`  ( R1 `  dom  z ) ) : ( R1
`  dom  z ) --> On )
5654, 15, 55syl2anc 643 . 2  |-  ( ph  ->  ( rank  |`  ( R1
`  dom  z )
) : ( R1
`  dom  z ) --> On )
57 epweon 4756 . . 3  |-  _E  We  On
5857a1i 11 . 2  |-  ( ph  ->  _E  We  On )
592, 3, 52, 56, 58fnwe2 27119 1  |-  ( ph  ->  F  We  ( R1
`  dom  z )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697   {crab 2701    C_ wss 3312   U.cuni 4007   class class class wbr 4204   {copab 4257    _E cep 4484    We wwe 4532   Ord word 4572   Oncon0 4573   suc csuc 4575   dom cdm 4870    |` cres 4872   "cima 4873   Fun wfun 5440    Fn wfn 5441   -->wf 5442   ` cfv 5446   R1cr1 7680   rankcrnk 7681
This theorem is referenced by:  aomclem5  27124
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-recs 6625  df-rdg 6660  df-r1 7682  df-rank 7683
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