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Theorem aomclem5 30979
Description: Lemma for dfac11 30983. 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 465 . . . . . 6  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  dom  z  e.  On )
4 simpr 461 . . . . . 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 465 . . . . . 6  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )
71, 3, 4, 6aomclem4 30978 . . . . 5  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  F  We  ( R1
`  dom  z )
)
8 iftrue 3932 . . . . . . 7  |-  ( dom  z  =  U. dom  z  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  =  F )
98adantl 466 . . . . . 6  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  =  F )
10 eqidd 2444 . . . . . 6  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  ( R1 `  dom  z )  =  ( R1 `  dom  z
) )
119, 10weeq12d 30960 . . . . 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 465 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  dom  z  e.  On )
18 eloni 4878 . . . . . . . 8  |-  ( dom  z  e.  On  ->  Ord 
dom  z )
19 orduniorsuc 6650 . . . . . . . 8  |-  ( Ord 
dom  z  ->  ( dom  z  =  U. dom  z  \/  dom  z  =  suc  U. dom  z ) )
202, 18, 193syl 20 . . . . . . 7  |-  ( ph  ->  ( dom  z  = 
U. dom  z  \/  dom  z  =  suc  U.
dom  z ) )
2120orcanai 913 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  dom  z  =  suc  U. dom  z )
225adantr 465 . . . . . 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 465 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  A  e.  On )
25 aomclem5.za . . . . . . 7  |-  ( ph  ->  dom  z  C_  A
)
2625adantr 465 . . . . . 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 465 . . . . . 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 30977 . . . . 5  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  E  We  ( R1 `  dom  z ) )
30 iffalse 3935 . . . . . . 7  |-  ( -. 
dom  z  =  U. dom  z  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  =  E )
3130adantl 466 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  =  E )
32 eqidd 2444 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  ( R1 `  dom  z )  =  ( R1 `  dom  z
) )
3331, 32weeq12d 30960 . . . . 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 791 . . 3  |-  ( ph  ->  if ( dom  z  =  U. dom  z ,  F ,  E )  We  ( R1 `  dom  z ) )
36 weinxp 5057 . . 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 4857 . . 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 368    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   E.wrex 2794   _Vcvv 3095    \ cdif 3458    i^i cin 3460    C_ wss 3461   (/)c0 3770   ifcif 3926   ~Pcpw 3997   {csn 4014   U.cuni 4234   |^|cint 4271   class class class wbr 4437   {copab 4494    |-> cmpt 4495    _E cep 4779    We wwe 4827   Ord word 4867   Oncon0 4868   suc csuc 4870    X. cxp 4987   `'ccnv 4988   dom cdm 4989   ran crn 4990   "cima 4992   ` cfv 5578  recscrecs 7043   Fincfn 7518   supcsup 7902   R1cr1 8183   rankcrnk 8184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-er 7313  df-map 7424  df-en 7519  df-fin 7522  df-sup 7903  df-r1 8185  df-rank 8186
This theorem is referenced by:  aomclem6  30980
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