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Theorem fpwwe2lem10 9017
Description: Lemma for fpwwe2 9021. 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 2467 . . . 4  |- OrdIso ( R ,  X )  = OrdIso
( R ,  X
)
21oicl 7954 . . 3  |-  Ord  dom OrdIso ( R ,  X )
3 eqid 2467 . . . 4  |- OrdIso ( S ,  Y )  = OrdIso
( S ,  Y
)
43oicl 7954 . . 3  |-  Ord  dom OrdIso ( S ,  Y )
5 ordtri2or2 4974 . . 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 9016 . . . 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 9016 . . . 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 836 . 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113   [.wsbc 3331    i^i cin 3475    C_ wss 3476   {csn 4027   class class class wbr 4447   {copab 4504    We wwe 4837   Ord word 4877    X. cxp 4997   `'ccnv 4998   dom cdm 4999   "cima 5002  (class class class)co 6284  OrdIsocoi 7934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-recs 7042  df-oi 7935
This theorem is referenced by:  fpwwe2lem11  9018  fpwwe2lem12  9019  fpwwe2  9021
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