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Theorem fpwwe2lem10 8277
Description: Lemma for fpwwe2 8281. 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 2296 . . . 4  |- OrdIso ( R ,  X )  = OrdIso
( R ,  X
)
21oicl 7260 . . 3  |-  Ord  dom OrdIso ( R ,  X )
3 eqid 2296 . . . 4  |- OrdIso ( S ,  Y )  = OrdIso
( S ,  Y
)
43oicl 7260 . . 3  |-  Ord  dom OrdIso ( S ,  Y )
5 ordtri2or2 4505 . . 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 8276 . . . 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 8276 . . . 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 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801   [.wsbc 3004    i^i cin 3164    C_ wss 3165   {csn 3653   class class class wbr 4039   {copab 4092    We wwe 4367   Ord word 4407    X. cxp 4703   `'ccnv 4704   dom cdm 4705   "cima 4708  (class class class)co 5874  OrdIsocoi 7240
This theorem is referenced by:  fpwwe2lem11  8278  fpwwe2lem12  8279  fpwwe2  8281
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-riota 6320  df-recs 6404  df-oi 7241
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