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Theorem aomclem5 27135
Description: Lemma for dfac11 27139. 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 453 . . . . . 6  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  dom  z  e.  On )
4 simpr 449 . . . . . 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 453 . . . . . 6  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )
71, 3, 4, 6aomclem4 27134 . . . . 5  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  F  We  ( R1
`  dom  z )
)
8 iftrue 3747 . . . . . . 7  |-  ( dom  z  =  U. dom  z  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  =  F )
98adantl 454 . . . . . 6  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  =  F )
10 eqidd 2439 . . . . . 6  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  ( R1 `  dom  z )  =  ( R1 `  dom  z
) )
119, 10weeq12d 27116 . . . . 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 225 . . . 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 453 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  dom  z  e.  On )
18 eloni 4593 . . . . . . . 8  |-  ( dom  z  e.  On  ->  Ord 
dom  z )
19 orduniorsuc 4812 . . . . . . . 8  |-  ( Ord 
dom  z  ->  ( dom  z  =  U. dom  z  \/  dom  z  =  suc  U. dom  z ) )
202, 18, 193syl 19 . . . . . . 7  |-  ( ph  ->  ( dom  z  = 
U. dom  z  \/  dom  z  =  suc  U.
dom  z ) )
2120orcanai 881 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  dom  z  =  suc  U. dom  z )
225adantr 453 . . . . . 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 453 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  A  e.  On )
25 aomclem5.za . . . . . . 7  |-  ( ph  ->  dom  z  C_  A
)
2625adantr 453 . . . . . 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 453 . . . . . 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 27133 . . . . 5  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  E  We  ( R1 `  dom  z ) )
30 iffalse 3748 . . . . . . 7  |-  ( -. 
dom  z  =  U. dom  z  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  =  E )
3130adantl 454 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  =  E )
32 eqidd 2439 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  ( R1 `  dom  z )  =  ( R1 `  dom  z
) )
3331, 32weeq12d 27116 . . . . 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 225 . . . 4  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  We  ( R1 `  dom  z ) )
3512, 34pm2.61dan 768 . . 3  |-  ( ph  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  We  ( R1 `  dom  z ) )
36 weinxp 4947 . . 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 190 . 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 4572 . . 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 8 . 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 205 1  |-  ( ph  ->  G  We  ( R1
`  dom  z )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   _Vcvv 2958    \ cdif 3319    i^i cin 3321    C_ wss 3322   (/)c0 3630   ifcif 3741   ~Pcpw 3801   {csn 3816   U.cuni 4017   |^|cint 4052   class class class wbr 4214   {copab 4267    e. cmpt 4268    _E cep 4494    We wwe 4542   Ord word 4582   Oncon0 4583   suc csuc 4585    X. cxp 4878   `'ccnv 4879   dom cdm 4880   ran crn 4881   "cima 4883   ` cfv 5456  recscrecs 6634   Fincfn 7111   supcsup 7447   R1cr1 7690   rankcrnk 7691
This theorem is referenced by:  aomclem6  27136
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-er 6907  df-map 7022  df-en 7112  df-fin 7115  df-sup 7448  df-r1 7692  df-rank 7693
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