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Theorem fpwwe2lem10 8261
Description: Lemma for fpwwe2 8265. 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 2283 . . . 4  |- OrdIso ( R ,  X )  = OrdIso
( R ,  X
)
21oicl 7244 . . 3  |-  Ord  dom OrdIso ( R ,  X )
3 eqid 2283 . . . 4  |- OrdIso ( S ,  Y )  = OrdIso
( S ,  Y
)
43oicl 7244 . . 3  |-  Ord  dom OrdIso ( S ,  Y )
5 ordtri2or2 4489 . . 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 653 . 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 451 . . . . 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 695 . . . . 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 451 . . . . 5  |-  ( (
ph  /\  dom OrdIso ( R ,  X )  C_  dom OrdIso ( S ,  Y
) )  ->  X W R )
14 fpwwe2lem10.6 . . . . . 6  |-  ( ph  ->  Y W S )
1514adantr 451 . . . . 5  |-  ( (
ph  /\  dom OrdIso ( R ,  X )  C_  dom OrdIso ( S ,  Y
) )  ->  Y W S )
16 simpr 447 . . . . 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 8260 . . . 4  |-  ( (
ph  /\  dom OrdIso ( R ,  X )  C_  dom OrdIso ( S ,  Y
) )  ->  ( X  C_  Y  /\  R  =  ( S  i^i  ( Y  X.  X
) ) ) )
1817ex 423 . . 3  |-  ( ph  ->  ( dom OrdIso ( R ,  X )  C_  dom OrdIso ( S ,  Y )  ->  ( X  C_  Y  /\  R  =  ( S  i^i  ( Y  X.  X ) ) ) ) )
198adantr 451 . . . . 5  |-  ( (
ph  /\  dom OrdIso ( S ,  Y )  C_  dom OrdIso ( R ,  X
) )  ->  A  e.  _V )
2010adantlr 695 . . . . 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 451 . . . . 5  |-  ( (
ph  /\  dom OrdIso ( S ,  Y )  C_  dom OrdIso ( R ,  X
) )  ->  Y W S )
2212adantr 451 . . . . 5  |-  ( (
ph  /\  dom OrdIso ( S ,  Y )  C_  dom OrdIso ( R ,  X
) )  ->  X W R )
23 simpr 447 . . . . 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 8260 . . . 4  |-  ( (
ph  /\  dom OrdIso ( S ,  Y )  C_  dom OrdIso ( R ,  X
) )  ->  ( Y  C_  X  /\  S  =  ( R  i^i  ( X  X.  Y
) ) ) )
2524ex 423 . . 3  |-  ( ph  ->  ( dom OrdIso ( S ,  Y )  C_  dom OrdIso ( R ,  X )  ->  ( Y  C_  X  /\  S  =  ( R  i^i  ( X  X.  Y ) ) ) ) )
2618, 25orim12d 811 . 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 16 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 set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   [.wsbc 2991    i^i cin 3151    C_ wss 3152   {csn 3640   class class class wbr 4023   {copab 4076    We wwe 4351   Ord word 4391    X. cxp 4687   `'ccnv 4688   dom cdm 4689   "cima 4692  (class class class)co 5858  OrdIsocoi 7224
This theorem is referenced by:  fpwwe2lem11  8262  fpwwe2lem12  8263  fpwwe2  8265
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-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-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  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-riota 6304  df-recs 6388  df-oi 7225
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