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Theorem aomclem4 35915
Description: Lemma for dfac11 35920. Limit case. Patch together well-orderings constructed so far using fnwe2 35911 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 5488 . . 3  |-  ( c  =  ( rank `  a
)  ->  suc  c  =  suc  ( rank `  a
) )
21fveq2d 5869 . 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 8238 . . . . . . . . . . . . . 14  |-  R1  Fn  On
5 fnfun 5673 . . . . . . . . . . . . . 14  |-  ( R1  Fn  On  ->  Fun  R1 )
64, 5ax-mp 5 . . . . . . . . . . . . 13  |-  Fun  R1
7 fndm 5675 . . . . . . . . . . . . . . 15  |-  ( R1  Fn  On  ->  dom  R1  =  On )
84, 7ax-mp 5 . . . . . . . . . . . . . 14  |-  dom  R1  =  On
98eqimss2i 3487 . . . . . . . . . . . . 13  |-  On  C_  dom  R1
106, 9pm3.2i 457 . . . . . . . . . . . 12  |-  ( Fun 
R1  /\  On  C_  dom  R1 )
11 aomclem4.on . . . . . . . . . . . 12  |-  ( ph  ->  dom  z  e.  On )
12 funfvima2 6141 . . . . . . . . . . . 12  |-  ( ( Fun  R1  /\  On  C_ 
dom  R1 )  ->  ( dom  z  e.  On  ->  ( R1 `  dom  z )  e.  ( R1 " On ) ) )
1310, 11, 12mpsyl 65 . . . . . . . . . . 11  |-  ( ph  ->  ( R1 `  dom  z )  e.  ( R1 " On ) )
14 elssuni 4227 . . . . . . . . . . 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 3432 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  ( R1 `  dom  z
) )  ->  b  e.  U. ( R1 " On ) )
17 rankidb 8271 . . . . . . . . 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 5488 . . . . . . . . . 10  |-  ( (
rank `  b )  =  ( rank `  a
)  ->  suc  ( rank `  b )  =  suc  ( rank `  a )
)
2019fveq2d 5869 . . . . . . . . 9  |-  ( (
rank `  b )  =  ( rank `  a
)  ->  ( R1 ` 
suc  ( rank `  b
) )  =  ( R1 `  suc  ( rank `  a ) ) )
2120eleq2d 2514 . . . . . . . 8  |-  ( (
rank `  b )  =  ( rank `  a
)  ->  ( b  e.  ( R1 `  suc  ( rank `  b )
)  <->  b  e.  ( R1 `  suc  ( rank `  a ) ) ) )
2218, 21syl5ibcom 224 . . . . . . 7  |-  ( (
ph  /\  b  e.  ( R1 `  dom  z
) )  ->  (
( rank `  b )  =  ( rank `  a
)  ->  b  e.  ( R1 `  suc  ( rank `  a ) ) ) )
2322expimpd 608 . . . . . 6  |-  ( ph  ->  ( ( b  e.  ( R1 `  dom  z )  /\  ( rank `  b )  =  ( rank `  a
) )  ->  b  e.  ( R1 `  suc  ( rank `  a )
) ) )
2423ss2abdv 3502 . . . . 5  |-  ( ph  ->  { b  |  ( b  e.  ( R1
`  dom  z )  /\  ( rank `  b
)  =  ( rank `  a ) ) } 
C_  { b  |  b  e.  ( R1
`  suc  ( rank `  a ) ) } )
25 df-rab 2746 . . . . 5  |-  { b  e.  ( R1 `  dom  z )  |  (
rank `  b )  =  ( rank `  a
) }  =  {
b  |  ( b  e.  ( R1 `  dom  z )  /\  ( rank `  b )  =  ( rank `  a
) ) }
26 abid1 2572 . . . . 5  |-  ( R1
`  suc  ( rank `  a ) )  =  { b  |  b  e.  ( R1 `  suc  ( rank `  a
) ) }
2724, 25, 263sstr4g 3473 . . . 4  |-  ( ph  ->  { b  e.  ( R1 `  dom  z
)  |  ( rank `  b )  =  (
rank `  a ) }  C_  ( R1 `  suc  ( rank `  a
) ) )
2827adantr 467 . . 3  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  { b  e.  ( R1 `  dom  z )  |  (
rank `  b )  =  ( rank `  a
) }  C_  ( R1 `  suc  ( rank `  a ) ) )
29 rankr1ai 8269 . . . . . 6  |-  ( a  e.  ( R1 `  dom  z )  ->  ( rank `  a )  e. 
dom  z )
3029adantl 468 . . . . 5  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  ( rank `  a )  e. 
dom  z )
31 eloni 5433 . . . . . . . 8  |-  ( dom  z  e.  On  ->  Ord 
dom  z )
3211, 31syl 17 . . . . . . 7  |-  ( ph  ->  Ord  dom  z )
33 aomclem4.su . . . . . . 7  |-  ( ph  ->  dom  z  =  U. dom  z )
34 limsuc2 35899 . . . . . . 7  |-  ( ( Ord  dom  z  /\  dom  z  =  U. dom  z )  ->  (
( rank `  a )  e.  dom  z  <->  suc  ( rank `  a )  e.  dom  z ) )
3532, 33, 34syl2anc 667 . . . . . 6  |-  ( ph  ->  ( ( rank `  a
)  e.  dom  z  <->  suc  ( rank `  a
)  e.  dom  z
) )
3635adantr 467 . . . . 5  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  (
( rank `  a )  e.  dom  z  <->  suc  ( rank `  a )  e.  dom  z ) )
3730, 36mpbid 214 . . . 4  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  suc  ( rank `  a )  e.  dom  z )
38 aomclem4.we . . . . . 6  |-  ( ph  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )
39 fveq2 5865 . . . . . . . 8  |-  ( a  =  b  ->  (
z `  a )  =  ( z `  b ) )
40 fveq2 5865 . . . . . . . 8  |-  ( a  =  b  ->  ( R1 `  a )  =  ( R1 `  b
) )
4139, 40weeq12d 35898 . . . . . . 7  |-  ( a  =  b  ->  (
( z `  a
)  We  ( R1
`  a )  <->  ( z `  b )  We  ( R1 `  b ) ) )
4241cbvralv 3019 . . . . . 6  |-  ( A. a  e.  dom  z ( z `  a )  We  ( R1 `  a )  <->  A. b  e.  dom  z ( z `
 b )  We  ( R1 `  b
) )
4338, 42sylib 200 . . . . 5  |-  ( ph  ->  A. b  e.  dom  z ( z `  b )  We  ( R1 `  b ) )
4443adantr 467 . . . 4  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  A. b  e.  dom  z ( z `
 b )  We  ( R1 `  b
) )
45 fveq2 5865 . . . . . 6  |-  ( b  =  suc  ( rank `  a )  ->  (
z `  b )  =  ( z `  suc  ( rank `  a
) ) )
46 fveq2 5865 . . . . . 6  |-  ( b  =  suc  ( rank `  a )  ->  ( R1 `  b )  =  ( R1 `  suc  ( rank `  a )
) )
4745, 46weeq12d 35898 . . . . 5  |-  ( b  =  suc  ( rank `  a )  ->  (
( z `  b
)  We  ( R1
`  b )  <->  ( z `  suc  ( rank `  a
) )  We  ( R1 `  suc  ( rank `  a ) ) ) )
4847rspcva 3148 . . . 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 ) ) )
4937, 44, 48syl2anc 667 . . 3  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  (
z `  suc  ( rank `  a ) )  We  ( R1 `  suc  ( rank `  a )
) )
50 wess 4821 . . 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 ) } ) )
5128, 49, 50sylc 62 . 2  |-  ( (
ph  /\  a  e.  ( R1 `  dom  z
) )  ->  (
z `  suc  ( rank `  a ) )  We 
{ b  e.  ( R1 `  dom  z
)  |  ( rank `  b )  =  (
rank `  a ) } )
52 rankf 8265 . . . 4  |-  rank : U. ( R1 " On ) --> On
5352a1i 11 . . 3  |-  ( ph  ->  rank : U. ( R1 " On ) --> On )
5453, 15fssresd 5750 . 2  |-  ( ph  ->  ( rank  |`  ( R1
`  dom  z )
) : ( R1
`  dom  z ) --> On )
55 epweon 6610 . . 3  |-  _E  We  On
5655a1i 11 . 2  |-  ( ph  ->  _E  We  On )
572, 3, 51, 54, 56fnwe2 35911 1  |-  ( ph  ->  F  We  ( R1
`  dom  z )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1444    e. wcel 1887   {cab 2437   A.wral 2737   {crab 2741    C_ wss 3404   U.cuni 4198   class class class wbr 4402   {copab 4460    _E cep 4743    We wwe 4792   dom cdm 4834   "cima 4837   Ord word 5422   Oncon0 5423   suc csuc 5425   Fun wfun 5576    Fn wfn 5577   -->wf 5578   ` cfv 5582   R1cr1 8233   rankcrnk 8234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-r1 8235  df-rank 8236
This theorem is referenced by:  aomclem5  35916
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