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Theorem aomclem5 27155
Description: Lemma for dfac11 27160. 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 27154 . . . . 5  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  F  We  ( R1
`  dom  z )
)
8 iftrue 3571 . . . . . . 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 2284 . . . . . 6  |-  ( (
ph  /\  dom  z  = 
U. dom  z )  ->  ( R1 `  dom  z )  =  ( R1 `  dom  z
) )
119, 10weeq12d 27136 . . . . 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 4402 . . . . . . . 8  |-  ( dom  z  e.  On  ->  Ord 
dom  z )
19 orduniorsuc 4621 . . . . . . . 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 27153 . . . . 5  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  E  We  ( R1 `  dom  z ) )
30 iffalse 3572 . . . . . . 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 2284 . . . . . 6  |-  ( (
ph  /\  -.  dom  z  =  U. dom  z )  ->  ( R1 `  dom  z )  =  ( R1 `  dom  z
) )
3331, 32weeq12d 27136 . . . . 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 4757 . . 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 4381 . . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   ifcif 3565   ~Pcpw 3625   {csn 3640   U.cuni 3827   |^|cint 3862   class class class wbr 4023   {copab 4076    e. cmpt 4077    _E cep 4303    We wwe 4351   Ord word 4391   Oncon0 4392   suc csuc 4394    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692   ` cfv 5255  recscrecs 6387   Fincfn 6863   supcsup 7193   R1cr1 7434   rankcrnk 7435
This theorem is referenced by:  aomclem6  27156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-er 6660  df-map 6774  df-en 6864  df-fin 6867  df-sup 7194  df-r1 7436  df-rank 7437
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