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Theorem fpwwe2lem10 8506
Description: Lemma for fpwwe2 8510. 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 2435 . . . 4  |- OrdIso ( R ,  X )  = OrdIso
( R ,  X
)
21oicl 7490 . . 3  |-  Ord  dom OrdIso ( R ,  X )
3 eqid 2435 . . . 4  |- OrdIso ( S ,  Y )  = OrdIso
( S ,  Y
)
43oicl 7490 . . 3  |-  Ord  dom OrdIso ( S ,  Y )
5 ordtri2or2 4670 . . 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 654 . 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 452 . . . . 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 696 . . . . 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 452 . . . . 5  |-  ( (
ph  /\  dom OrdIso ( R ,  X )  C_  dom OrdIso ( S ,  Y
) )  ->  X W R )
14 fpwwe2lem10.6 . . . . . 6  |-  ( ph  ->  Y W S )
1514adantr 452 . . . . 5  |-  ( (
ph  /\  dom OrdIso ( R ,  X )  C_  dom OrdIso ( S ,  Y
) )  ->  Y W S )
16 simpr 448 . . . . 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 8505 . . . 4  |-  ( (
ph  /\  dom OrdIso ( R ,  X )  C_  dom OrdIso ( S ,  Y
) )  ->  ( X  C_  Y  /\  R  =  ( S  i^i  ( Y  X.  X
) ) ) )
1817ex 424 . . 3  |-  ( ph  ->  ( dom OrdIso ( R ,  X )  C_  dom OrdIso ( S ,  Y )  ->  ( X  C_  Y  /\  R  =  ( S  i^i  ( Y  X.  X ) ) ) ) )
198adantr 452 . . . . 5  |-  ( (
ph  /\  dom OrdIso ( S ,  Y )  C_  dom OrdIso ( R ,  X
) )  ->  A  e.  _V )
2010adantlr 696 . . . . 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 452 . . . . 5  |-  ( (
ph  /\  dom OrdIso ( S ,  Y )  C_  dom OrdIso ( R ,  X
) )  ->  Y W S )
2212adantr 452 . . . . 5  |-  ( (
ph  /\  dom OrdIso ( S ,  Y )  C_  dom OrdIso ( R ,  X
) )  ->  X W R )
23 simpr 448 . . . . 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 8505 . . . 4  |-  ( (
ph  /\  dom OrdIso ( S ,  Y )  C_  dom OrdIso ( R ,  X
) )  ->  ( Y  C_  X  /\  S  =  ( R  i^i  ( X  X.  Y
) ) ) )
2524ex 424 . . 3  |-  ( ph  ->  ( dom OrdIso ( S ,  Y )  C_  dom OrdIso ( R ,  X )  ->  ( Y  C_  X  /\  S  =  ( R  i^i  ( X  X.  Y ) ) ) ) )
2618, 25orim12d 812 . 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 set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948   [.wsbc 3153    i^i cin 3311    C_ wss 3312   {csn 3806   class class class wbr 4204   {copab 4257    We wwe 4532   Ord word 4572    X. cxp 4868   `'ccnv 4869   dom cdm 4870   "cima 4873  (class class class)co 6073  OrdIsocoi 7470
This theorem is referenced by:  fpwwe2lem11  8507  fpwwe2lem12  8508  fpwwe2  8510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-riota 6541  df-recs 6625  df-oi 7471
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