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Theorem aomclem4 35345
Description: Lemma for dfac11 35350. Limit case. Patch together well-orderings constructed so far using fnwe2 35341 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 5474 . . 3  |-  ( c  =  ( rank `  a
)  ->  suc  c  =  suc  ( rank `  a
) )
21fveq2d 5852 . 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 8216 . . . . . . . . . . . . . 14  |-  R1  Fn  On
5 fnfun 5658 . . . . . . . . . . . . . 14  |-  ( R1  Fn  On  ->  Fun  R1 )
64, 5ax-mp 5 . . . . . . . . . . . . 13  |-  Fun  R1
7 fndm 5660 . . . . . . . . . . . . . . 15  |-  ( R1  Fn  On  ->  dom  R1  =  On )
84, 7ax-mp 5 . . . . . . . . . . . . . 14  |-  dom  R1  =  On
98eqimss2i 3496 . . . . . . . . . . . . 13  |-  On  C_  dom  R1
106, 9pm3.2i 453 . . . . . . . . . . . 12  |-  ( Fun 
R1  /\  On  C_  dom  R1 )
11 aomclem4.on . . . . . . . . . . . 12  |-  ( ph  ->  dom  z  e.  On )
12 funfvima2 6128 . . . . . . . . . . . 12  |-  ( ( Fun  R1  /\  On  C_ 
dom  R1 )  ->  ( dom  z  e.  On  ->  ( R1 `  dom  z )  e.  ( R1 " On ) ) )
1310, 11, 12mpsyl 62 . . . . . . . . . . 11  |-  ( ph  ->  ( R1 `  dom  z )  e.  ( R1 " On ) )
14 elssuni 4219 . . . . . . . . . . 11  |-  ( ( R1 `  dom  z
)  e.  ( R1
" On )  -> 
( R1 `  dom  z )  C_  U. ( R1 " On ) )
1513, 14syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( R1 `  dom  z )  C_  U. ( R1 " On ) )
1615sselda 3441 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  ( R1 `  dom  z
) )  ->  b  e.  U. ( R1 " On ) )
17 rankidb 8249 . . . . . . . . 9  |-  ( b  e.  U. ( R1
" On )  -> 
b  e.  ( R1
`  suc  ( rank `  b ) ) )
1816, 17syl 17 . . . . . . . 8  |-  ( (
ph  /\  b  e.  ( R1 `  dom  z
) )  ->  b  e.  ( R1 `  suc  ( rank `  b )
) )
19 suceq 5474 . . . . . . . . . 10  |-  ( (
rank `  b )  =  ( rank `  a
)  ->  suc  ( rank `  b )  =  suc  ( rank `  a )
)
2019fveq2d 5852 . . . . . . . . 9  |-  ( (
rank `  b )  =  ( rank `  a
)  ->  ( R1 ` 
suc  ( rank `  b
) )  =  ( R1 `  suc  ( rank `  a ) ) )
2120eleq2d 2472 . . . . . . . 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 601 . . . . . 6  |-  ( ph  ->  ( ( b  e.  ( R1 `  dom  z )  /\  ( rank `  b )  =  ( rank `  a
) )  ->  b  e.  ( R1 `  suc  ( rank `  a )
) ) )
2423ss2abdv 3511 . . . . 5  |-  ( ph  ->  { b  |  ( b  e.  ( R1
`  dom  z )  /\  ( rank `  b
)  =  ( rank `  a ) ) } 
C_  { b  |  b  e.  ( R1
`  suc  ( rank `  a ) ) } )
25 df-rab 2762 . . . . 5  |-  { b  e.  ( R1 `  dom  z )  |  (
rank `  b )  =  ( rank `  a
) }  =  {
b  |  ( b  e.  ( R1 `  dom  z )  /\  ( rank `  b )  =  ( rank `  a
) ) }
26 abid2 2542 . . . . . 6  |-  { b  |  b  e.  ( R1 `  suc  ( rank `  a ) ) }  =  ( R1
`  suc  ( rank `  a ) )
2726eqcomi 2415 . . . . 5  |-  ( R1
`  suc  ( rank `  a ) )  =  { b  |  b  e.  ( R1 `  suc  ( rank `  a
) ) }
2824, 25, 273sstr4g 3482 . . . 4  |-  ( ph  ->  { b  e.  ( R1 `  dom  z
)  |  ( rank `  b )  =  (
rank `  a ) }  C_  ( R1 `  suc  ( rank `  a
) ) )
2928adantr 463 . . 3  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  { b  e.  ( R1 `  dom  z )  |  (
rank `  b )  =  ( rank `  a
) }  C_  ( R1 `  suc  ( rank `  a ) ) )
30 rankr1ai 8247 . . . . . 6  |-  ( a  e.  ( R1 `  dom  z )  ->  ( rank `  a )  e. 
dom  z )
3130adantl 464 . . . . 5  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  ( rank `  a )  e. 
dom  z )
32 eloni 5419 . . . . . . . 8  |-  ( dom  z  e.  On  ->  Ord 
dom  z )
3311, 32syl 17 . . . . . . 7  |-  ( ph  ->  Ord  dom  z )
34 aomclem4.su . . . . . . 7  |-  ( ph  ->  dom  z  =  U. dom  z )
35 limsuc2 35328 . . . . . . 7  |-  ( ( Ord  dom  z  /\  dom  z  =  U. dom  z )  ->  (
( rank `  a )  e.  dom  z  <->  suc  ( rank `  a )  e.  dom  z ) )
3633, 34, 35syl2anc 659 . . . . . 6  |-  ( ph  ->  ( ( rank `  a
)  e.  dom  z  <->  suc  ( rank `  a
)  e.  dom  z
) )
3736adantr 463 . . . . 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 5848 . . . . . . . 8  |-  ( a  =  b  ->  (
z `  a )  =  ( z `  b ) )
41 fveq2 5848 . . . . . . . 8  |-  ( a  =  b  ->  ( R1 `  a )  =  ( R1 `  b
) )
4240, 41weeq12d 35327 . . . . . . 7  |-  ( a  =  b  ->  (
( z `  a
)  We  ( R1
`  a )  <->  ( z `  b )  We  ( R1 `  b ) ) )
4342cbvralv 3033 . . . . . 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 463 . . . 4  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  A. b  e.  dom  z ( z `
 b )  We  ( R1 `  b
) )
46 fveq2 5848 . . . . . 6  |-  ( b  =  suc  ( rank `  a )  ->  (
z `  b )  =  ( z `  suc  ( rank `  a
) ) )
47 fveq2 5848 . . . . . 6  |-  ( b  =  suc  ( rank `  a )  ->  ( R1 `  b )  =  ( R1 `  suc  ( rank `  a )
) )
4846, 47weeq12d 35327 . . . . 5  |-  ( b  =  suc  ( rank `  a )  ->  (
( z `  b
)  We  ( R1
`  b )  <->  ( z `  suc  ( rank `  a
) )  We  ( R1 `  suc  ( rank `  a ) ) ) )
4948rspcva 3157 . . . 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 659 . . 3  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  (
z `  suc  ( rank `  a ) )  We  ( R1 `  suc  ( rank `  a )
) )
51 wess 4809 . . 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 59 . 2  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  (
z `  suc  ( rank `  a ) )  We 
{ b  e.  ( R1 `  dom  z
)  |  ( rank `  b )  =  (
rank `  a ) } )
53 rankf 8243 . . . 4  |-  rank : U. ( R1 " On ) --> On
5453a1i 11 . . 3  |-  ( ph  ->  rank : U. ( R1 " On ) --> On )
5554, 15fssresd 5734 . 2  |-  ( ph  ->  ( rank  |`  ( R1
`  dom  z )
) : ( R1
`  dom  z ) --> On )
56 epweon 6600 . . 3  |-  _E  We  On
5756a1i 11 . 2  |-  ( ph  ->  _E  We  On )
582, 3, 52, 55, 57fnwe2 35341 1  |-  ( ph  ->  F  We  ( R1
`  dom  z )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842   {cab 2387   A.wral 2753   {crab 2757    C_ wss 3413   U.cuni 4190   class class class wbr 4394   {copab 4451    _E cep 4731    We wwe 4780   dom cdm 4822   "cima 4825   Ord word 5408   Oncon0 5409   suc csuc 5411   Fun wfun 5562    Fn wfn 5563   -->wf 5564   ` cfv 5568   R1cr1 8211   rankcrnk 8212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-om 6683  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-r1 8213  df-rank 8214
This theorem is referenced by:  aomclem5  35346
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