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Theorem fpwwe2lem10 8805
Description: Lemma for fpwwe2 8809. Given two well-orders  <. X ,  R >. and  <. Y ,  S >. of parts of  A, one is an initial segment of the other. (Contributed by Mario Carneiro, 15-May-2015.)
Hypotheses
Ref Expression
fpwwe2.1  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. (
u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }
fpwwe2.2  |-  ( ph  ->  A  e.  _V )
fpwwe2.3  |-  ( (
ph  /\  ( x  C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x ) )  -> 
( x F r )  e.  A )
fpwwe2lem10.4  |-  ( ph  ->  X W R )
fpwwe2lem10.6  |-  ( ph  ->  Y W S )
Assertion
Ref Expression
fpwwe2lem10  |-  ( ph  ->  ( ( X  C_  Y  /\  R  =  ( S  i^i  ( Y  X.  X ) ) )  \/  ( Y 
C_  X  /\  S  =  ( R  i^i  ( X  X.  Y
) ) ) ) )
Distinct variable groups:    y, u, r, x, F    X, r, u, x, y    ph, r, u, x, y    A, r, x    R, r, u, x, y    Y, r, u, x, y    S, r, u, x, y    W, r, u, x, y
Allowed substitution hints:    A( y, u)

Proof of Theorem fpwwe2lem10
StepHypRef Expression
1 eqid 2442 . . . 4  |- OrdIso ( R ,  X )  = OrdIso
( R ,  X
)
21oicl 7742 . . 3  |-  Ord  dom OrdIso ( R ,  X )
3 eqid 2442 . . . 4  |- OrdIso ( S ,  Y )  = OrdIso
( S ,  Y
)
43oicl 7742 . . 3  |-  Ord  dom OrdIso ( S ,  Y )
5 ordtri2or2 4814 . . 3  |-  ( ( Ord  dom OrdIso ( R ,  X )  /\  Ord  dom OrdIso ( S ,  Y ) )  ->  ( dom OrdIso ( R ,  X ) 
C_  dom OrdIso ( S ,  Y )  \/  dom OrdIso ( S ,  Y ) 
C_  dom OrdIso ( R ,  X ) ) )
62, 4, 5mp2an 672 . 2  |-  ( dom OrdIso ( R ,  X ) 
C_  dom OrdIso ( S ,  Y )  \/  dom OrdIso ( S ,  Y ) 
C_  dom OrdIso ( R ,  X ) )
7 fpwwe2.1 . . . . 5  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. (
u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }
8 fpwwe2.2 . . . . . 6  |-  ( ph  ->  A  e.  _V )
98adantr 465 . . . . 5  |-  ( (
ph  /\  dom OrdIso ( R ,  X )  C_  dom OrdIso ( S ,  Y
) )  ->  A  e.  _V )
10 fpwwe2.3 . . . . . 6  |-  ( (
ph  /\  ( x  C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x ) )  -> 
( x F r )  e.  A )
1110adantlr 714 . . . . 5  |-  ( ( ( ph  /\  dom OrdIso ( R ,  X ) 
C_  dom OrdIso ( S ,  Y ) )  /\  ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) )  -> 
( x F r )  e.  A )
12 fpwwe2lem10.4 . . . . . 6  |-  ( ph  ->  X W R )
1312adantr 465 . . . . 5  |-  ( (
ph  /\  dom OrdIso ( R ,  X )  C_  dom OrdIso ( S ,  Y
) )  ->  X W R )
14 fpwwe2lem10.6 . . . . . 6  |-  ( ph  ->  Y W S )
1514adantr 465 . . . . 5  |-  ( (
ph  /\  dom OrdIso ( R ,  X )  C_  dom OrdIso ( S ,  Y
) )  ->  Y W S )
16 simpr 461 . . . . 5  |-  ( (
ph  /\  dom OrdIso ( R ,  X )  C_  dom OrdIso ( S ,  Y
) )  ->  dom OrdIso ( R ,  X ) 
C_  dom OrdIso ( S ,  Y ) )
177, 9, 11, 13, 15, 1, 3, 16fpwwe2lem9 8804 . . . 4  |-  ( (
ph  /\  dom OrdIso ( R ,  X )  C_  dom OrdIso ( S ,  Y
) )  ->  ( X  C_  Y  /\  R  =  ( S  i^i  ( Y  X.  X
) ) ) )
1817ex 434 . . 3  |-  ( ph  ->  ( dom OrdIso ( R ,  X )  C_  dom OrdIso ( S ,  Y )  ->  ( X  C_  Y  /\  R  =  ( S  i^i  ( Y  X.  X ) ) ) ) )
198adantr 465 . . . . 5  |-  ( (
ph  /\  dom OrdIso ( S ,  Y )  C_  dom OrdIso ( R ,  X
) )  ->  A  e.  _V )
2010adantlr 714 . . . . 5  |-  ( ( ( ph  /\  dom OrdIso ( S ,  Y ) 
C_  dom OrdIso ( R ,  X ) )  /\  ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) )  -> 
( x F r )  e.  A )
2114adantr 465 . . . . 5  |-  ( (
ph  /\  dom OrdIso ( S ,  Y )  C_  dom OrdIso ( R ,  X
) )  ->  Y W S )
2212adantr 465 . . . . 5  |-  ( (
ph  /\  dom OrdIso ( S ,  Y )  C_  dom OrdIso ( R ,  X
) )  ->  X W R )
23 simpr 461 . . . . 5  |-  ( (
ph  /\  dom OrdIso ( S ,  Y )  C_  dom OrdIso ( R ,  X
) )  ->  dom OrdIso ( S ,  Y ) 
C_  dom OrdIso ( R ,  X ) )
247, 19, 20, 21, 22, 3, 1, 23fpwwe2lem9 8804 . . . 4  |-  ( (
ph  /\  dom OrdIso ( S ,  Y )  C_  dom OrdIso ( R ,  X
) )  ->  ( Y  C_  X  /\  S  =  ( R  i^i  ( X  X.  Y
) ) ) )
2524ex 434 . . 3  |-  ( ph  ->  ( dom OrdIso ( S ,  Y )  C_  dom OrdIso ( R ,  X )  ->  ( Y  C_  X  /\  S  =  ( R  i^i  ( X  X.  Y ) ) ) ) )
2618, 25orim12d 834 . 2  |-  ( ph  ->  ( ( dom OrdIso ( R ,  X )  C_  dom OrdIso ( S ,  Y
)  \/  dom OrdIso ( S ,  Y )  C_  dom OrdIso ( R ,  X
) )  ->  (
( X  C_  Y  /\  R  =  ( S  i^i  ( Y  X.  X ) ) )  \/  ( Y  C_  X  /\  S  =  ( R  i^i  ( X  X.  Y ) ) ) ) ) )
276, 26mpi 17 1  |-  ( ph  ->  ( ( X  C_  Y  /\  R  =  ( S  i^i  ( Y  X.  X ) ) )  \/  ( Y 
C_  X  /\  S  =  ( R  i^i  ( X  X.  Y
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2714   _Vcvv 2971   [.wsbc 3185    i^i cin 3326    C_ wss 3327   {csn 3876   class class class wbr 4291   {copab 4348    We wwe 4677   Ord word 4717    X. cxp 4837   `'ccnv 4838   dom cdm 4839   "cima 4842  (class class class)co 6090  OrdIsocoi 7722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-recs 6831  df-oi 7723
This theorem is referenced by:  fpwwe2lem11  8806  fpwwe2lem12  8807  fpwwe2  8809
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