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Theorem aomclem5 35366
Description: Lemma for dfac11 35370. Combine the successor case with the limit case. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Hypotheses
Ref Expression
aomclem5.b  |-  B  =  { <. a ,  b
>.  |  E. c  e.  ( R1 `  U. dom  z ) ( ( c  e.  b  /\  -.  c  e.  a
)  /\  A. d  e.  ( R1 `  U. dom  z ) ( d ( z `  U. dom  z ) c  -> 
( d  e.  a  <-> 
d  e.  b ) ) ) }
aomclem5.c  |-  C  =  ( a  e.  _V  |->  sup ( ( y `  a ) ,  ( R1 `  dom  z
) ,  B ) )
aomclem5.d  |-  D  = recs ( ( a  e. 
_V  |->  ( C `  ( ( R1 `  dom  z )  \  ran  a ) ) ) )
aomclem5.e  |-  E  =  { <. a ,  b
>.  |  |^| ( `' D " { a } )  e.  |^| ( `' D " { b } ) }
aomclem5.f  |-  F  =  { <. a ,  b
>.  |  ( ( rank `  a )  _E  ( rank `  b
)  \/  ( (
rank `  a )  =  ( rank `  b
)  /\  a (
z `  suc  ( rank `  a ) ) b ) ) }
aomclem5.g  |-  G  =  ( if ( dom  z  =  U. dom  z ,  F ,  E )  i^i  (
( R1 `  dom  z )  X.  ( R1 `  dom  z ) ) )
aomclem5.on  |-  ( ph  ->  dom  z  e.  On )
aomclem5.we  |-  ( ph  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )
aomclem5.a  |-  ( ph  ->  A  e.  On )
aomclem5.za  |-  ( ph  ->  dom  z  C_  A
)
aomclem5.y  |-  ( ph  ->  A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  (
y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
} ) ) )
Assertion
Ref Expression
aomclem5  |-  ( ph  ->  G  We  ( R1
`  dom  z )
)
Distinct variable groups:    y, z,
a, b, c, d    ph, a, b    C, a, b, c, d    D, a, b, c, d
Allowed substitution hints:    ph( y, z, c, d)    A( y, z, a, b, c, d)    B( y, z, a, b, c, d)    C( y, z)    D( y, z)    E( y, z, a, b, c, d)    F( y, z, a, b, c, d)    G( y, z, a, b, c, d)

Proof of Theorem aomclem5
StepHypRef Expression
1 aomclem5.f . . . . . 6  |-  F  =  { <. a ,  b
>.  |  ( ( rank `  a )  _E  ( rank `  b
)  \/  ( (
rank `  a )  =  ( rank `  b
)  /\  a (
z `  suc  ( rank `  a ) ) b ) ) }
2 aomclem5.on . . . . . . 7  |-  ( ph  ->  dom  z  e.  On )
32adantr 463 . . . . . 6  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  dom  z  e.  On )
4 simpr 459 . . . . . 6  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  dom  z  =  U. dom  z )
5 aomclem5.we . . . . . . 7  |-  ( ph  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )
65adantr 463 . . . . . 6  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )
71, 3, 4, 6aomclem4 35365 . . . . 5  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  F  We  ( R1
`  dom  z )
)
8 iftrue 3891 . . . . . . 7  |-  ( dom  z  =  U. dom  z  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  =  F )
98adantl 464 . . . . . 6  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  =  F )
10 eqidd 2403 . . . . . 6  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  ( R1 `  dom  z )  =  ( R1 `  dom  z
) )
119, 10weeq12d 35347 . . . . 5  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  ( if ( dom  z  =  U. dom  z ,  F ,  E )  We  ( R1 `  dom  z )  <-> 
F  We  ( R1
`  dom  z )
) )
127, 11mpbird 232 . . . 4  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  We  ( R1 `  dom  z ) )
13 aomclem5.b . . . . . 6  |-  B  =  { <. a ,  b
>.  |  E. c  e.  ( R1 `  U. dom  z ) ( ( c  e.  b  /\  -.  c  e.  a
)  /\  A. d  e.  ( R1 `  U. dom  z ) ( d ( z `  U. dom  z ) c  -> 
( d  e.  a  <-> 
d  e.  b ) ) ) }
14 aomclem5.c . . . . . 6  |-  C  =  ( a  e.  _V  |->  sup ( ( y `  a ) ,  ( R1 `  dom  z
) ,  B ) )
15 aomclem5.d . . . . . 6  |-  D  = recs ( ( a  e. 
_V  |->  ( C `  ( ( R1 `  dom  z )  \  ran  a ) ) ) )
16 aomclem5.e . . . . . 6  |-  E  =  { <. a ,  b
>.  |  |^| ( `' D " { a } )  e.  |^| ( `' D " { b } ) }
172adantr 463 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  dom  z  e.  On )
18 eloni 5420 . . . . . . . 8  |-  ( dom  z  e.  On  ->  Ord 
dom  z )
19 orduniorsuc 6648 . . . . . . . 8  |-  ( Ord 
dom  z  ->  ( dom  z  =  U. dom  z  \/  dom  z  =  suc  U. dom  z ) )
202, 18, 193syl 18 . . . . . . 7  |-  ( ph  ->  ( dom  z  = 
U. dom  z  \/  dom  z  =  suc  U.
dom  z ) )
2120orcanai 914 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  dom  z  =  suc  U. dom  z )
225adantr 463 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )
23 aomclem5.a . . . . . . 7  |-  ( ph  ->  A  e.  On )
2423adantr 463 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  A  e.  On )
25 aomclem5.za . . . . . . 7  |-  ( ph  ->  dom  z  C_  A
)
2625adantr 463 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  dom  z  C_  A )
27 aomclem5.y . . . . . . 7  |-  ( ph  ->  A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  (
y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
} ) ) )
2827adantr 463 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  (
y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
} ) ) )
2913, 14, 15, 16, 17, 21, 22, 24, 26, 28aomclem3 35364 . . . . 5  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  E  We  ( R1 `  dom  z ) )
30 iffalse 3894 . . . . . . 7  |-  ( -. 
dom  z  =  U. dom  z  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  =  E )
3130adantl 464 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  =  E )
32 eqidd 2403 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  ( R1 `  dom  z )  =  ( R1 `  dom  z
) )
3331, 32weeq12d 35347 . . . . 5  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  ( if ( dom  z  =  U. dom  z ,  F ,  E )  We  ( R1 `  dom  z )  <-> 
E  We  ( R1
`  dom  z )
) )
3429, 33mpbird 232 . . . 4  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  We  ( R1 `  dom  z ) )
3512, 34pm2.61dan 792 . . 3  |-  ( ph  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  We  ( R1 `  dom  z ) )
36 weinxp 4891 . . 3  |-  ( if ( dom  z  = 
U. dom  z ,  F ,  E )  We  ( R1 `  dom  z )  <->  ( if ( dom  z  =  U. dom  z ,  F ,  E )  i^i  (
( R1 `  dom  z )  X.  ( R1 `  dom  z ) ) )  We  ( R1 `  dom  z ) )
3735, 36sylib 196 . 2  |-  ( ph  ->  ( if ( dom  z  =  U. dom  z ,  F ,  E )  i^i  (
( R1 `  dom  z )  X.  ( R1 `  dom  z ) ) )  We  ( R1 `  dom  z ) )
38 aomclem5.g . . 3  |-  G  =  ( if ( dom  z  =  U. dom  z ,  F ,  E )  i^i  (
( R1 `  dom  z )  X.  ( R1 `  dom  z ) ) )
39 weeq1 4811 . . 3  |-  ( G  =  ( if ( dom  z  =  U. dom  z ,  F ,  E )  i^i  (
( R1 `  dom  z )  X.  ( R1 `  dom  z ) ) )  ->  ( G  We  ( R1 ` 
dom  z )  <->  ( if ( dom  z  =  U. dom  z ,  F ,  E )  i^i  (
( R1 `  dom  z )  X.  ( R1 `  dom  z ) ) )  We  ( R1 `  dom  z ) ) )
4038, 39ax-mp 5 . 2  |-  ( G  We  ( R1 `  dom  z )  <->  ( if ( dom  z  =  U. dom  z ,  F ,  E )  i^i  (
( R1 `  dom  z )  X.  ( R1 `  dom  z ) ) )  We  ( R1 `  dom  z ) )
4137, 40sylibr 212 1  |-  ( ph  ->  G  We  ( R1
`  dom  z )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   E.wrex 2755   _Vcvv 3059    \ cdif 3411    i^i cin 3413    C_ wss 3414   (/)c0 3738   ifcif 3885   ~Pcpw 3955   {csn 3972   U.cuni 4191   |^|cint 4227   class class class wbr 4395   {copab 4452    |-> cmpt 4453    _E cep 4732    We wwe 4781    X. cxp 4821   `'ccnv 4822   dom cdm 4823   ran crn 4824   "cima 4826   Ord word 5409   Oncon0 5410   suc csuc 5412   ` cfv 5569  recscrecs 7074   Fincfn 7554   supcsup 7934   R1cr1 8212   rankcrnk 8213
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 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
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 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-er 7348  df-map 7459  df-en 7555  df-fin 7558  df-sup 7935  df-r1 8214  df-rank 8215
This theorem is referenced by:  aomclem6  35367
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