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Theorem aomclem5 27258
Description: Lemma for dfac11 27263. 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 451 . . . . . 6  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  dom  z  e.  On )
4 simpr 447 . . . . . 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 451 . . . . . 6  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )
71, 3, 4, 6aomclem4 27257 . . . . 5  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  F  We  ( R1
`  dom  z )
)
8 iftrue 3584 . . . . . . 7  |-  ( dom  z  =  U. dom  z  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  =  F )
98adantl 452 . . . . . 6  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  =  F )
10 eqidd 2297 . . . . . 6  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  ( R1 `  dom  z )  =  ( R1 `  dom  z
) )
119, 10weeq12d 27239 . . . . 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 223 . . . 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 451 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  dom  z  e.  On )
18 eloni 4418 . . . . . . . 8  |-  ( dom  z  e.  On  ->  Ord 
dom  z )
19 orduniorsuc 4637 . . . . . . . 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 879 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  dom  z  =  suc  U. dom  z )
225adantr 451 . . . . . 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 451 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  A  e.  On )
25 aomclem5.za . . . . . . 7  |-  ( ph  ->  dom  z  C_  A
)
2625adantr 451 . . . . . 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 451 . . . . . 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 27256 . . . . 5  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  E  We  ( R1 `  dom  z ) )
30 iffalse 3585 . . . . . . 7  |-  ( -. 
dom  z  =  U. dom  z  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  =  E )
3130adantl 452 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  =  E )
32 eqidd 2297 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  ( R1 `  dom  z )  =  ( R1 `  dom  z
) )
3331, 32weeq12d 27239 . . . . 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 223 . . . 4  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  We  ( R1 `  dom  z ) )
3512, 34pm2.61dan 766 . . 3  |-  ( ph  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  We  ( R1 `  dom  z ) )
36 weinxp 4773 . . 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 188 . 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 4397 . . 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 203 1  |-  ( ph  ->  G  We  ( R1
`  dom  z )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   ifcif 3578   ~Pcpw 3638   {csn 3653   U.cuni 3843   |^|cint 3878   class class class wbr 4039   {copab 4092    e. cmpt 4093    _E cep 4319    We wwe 4367   Ord word 4407   Oncon0 4408   suc csuc 4410    X. cxp 4703   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708   ` cfv 5271  recscrecs 6403   Fincfn 6879   supcsup 7209   R1cr1 7450   rankcrnk 7451
This theorem is referenced by:  aomclem6  27259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-er 6676  df-map 6790  df-en 6880  df-fin 6883  df-sup 7210  df-r1 7452  df-rank 7453
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