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Theorem fnwe2 27150
Description: A well-ordering can be constructed on a partitioned set by patching together well-orderings on each partition using a well-ordering on the partitions themselves. Similar to fnwe 6231 but does not require the within-partition ordering to be globally well. (Contributed by Stefan O'Rear, 19-Jan-2015.)
Hypotheses
Ref Expression
fnwe2.su  |-  ( z  =  ( F `  x )  ->  S  =  U )
fnwe2.t  |-  T  =  { <. x ,  y
>.  |  ( ( F `  x ) R ( F `  y )  \/  (
( F `  x
)  =  ( F `
 y )  /\  x U y ) ) }
fnwe2.s  |-  ( (
ph  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
fnwe2.f  |-  ( ph  ->  ( F  |`  A ) : A --> B )
fnwe2.r  |-  ( ph  ->  R  We  B )
Assertion
Ref Expression
fnwe2  |-  ( ph  ->  T  We  A )
Distinct variable groups:    y, U, z    x, S, y    x, R, y    ph, x, y, z    x, A, y, z    x, F, y, z
Allowed substitution hints:    B( x, y, z)    R( z)    S( z)    T( x, y, z)    U( x)

Proof of Theorem fnwe2
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnwe2.su . . . . . 6  |-  ( z  =  ( F `  x )  ->  S  =  U )
2 fnwe2.t . . . . . 6  |-  T  =  { <. x ,  y
>.  |  ( ( F `  x ) R ( F `  y )  \/  (
( F `  x
)  =  ( F `
 y )  /\  x U y ) ) }
3 fnwe2.s . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
43adantlr 695 . . . . . 6  |-  ( ( ( ph  /\  (
a  C_  A  /\  a  =/=  (/) ) )  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
5 fnwe2.f . . . . . . 7  |-  ( ph  ->  ( F  |`  A ) : A --> B )
65adantr 451 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  ( F  |`  A ) : A --> B )
7 fnwe2.r . . . . . . 7  |-  ( ph  ->  R  We  B )
87adantr 451 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  R  We  B )
9 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  a  C_  A )
10 simprr 733 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  a  =/=  (/) )
111, 2, 4, 6, 8, 9, 10fnwe2lem2 27148 . . . . 5  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  E. c  e.  a  A. d  e.  a  -.  d T c )
1211ex 423 . . . 4  |-  ( ph  ->  ( ( a  C_  A  /\  a  =/=  (/) )  ->  E. c  e.  a  A. d  e.  a  -.  d T c ) )
1312alrimiv 1617 . . 3  |-  ( ph  ->  A. a ( ( a  C_  A  /\  a  =/=  (/) )  ->  E. c  e.  a  A. d  e.  a  -.  d T c ) )
14 df-fr 4352 . . 3  |-  ( T  Fr  A  <->  A. a
( ( a  C_  A  /\  a  =/=  (/) )  ->  E. c  e.  a  A. d  e.  a  -.  d T c ) )
1513, 14sylibr 203 . 2  |-  ( ph  ->  T  Fr  A )
163adantlr 695 . . . 4  |-  ( ( ( ph  /\  (
a  e.  A  /\  b  e.  A )
)  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
175adantr 451 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
( F  |`  A ) : A --> B )
187adantr 451 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  ->  R  We  B )
19 simprl 732 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
a  e.  A )
20 simprr 733 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
b  e.  A )
211, 2, 16, 17, 18, 19, 20fnwe2lem3 27149 . . 3  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
( a T b  \/  a  =  b  \/  b T a ) )
2221ralrimivva 2635 . 2  |-  ( ph  ->  A. a  e.  A  A. b  e.  A  ( a T b  \/  a  =  b  \/  b T a ) )
23 dfwe2 4573 . 2  |-  ( T  We  A  <->  ( T  Fr  A  /\  A. a  e.  A  A. b  e.  A  ( a T b  \/  a  =  b  \/  b T a ) ) )
2415, 22, 23sylanbrc 645 1  |-  ( ph  ->  T  We  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    \/ w3o 933   A.wal 1527    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547    C_ wss 3152   (/)c0 3455   class class class wbr 4023   {copab 4076    Fr wfr 4349    We wwe 4351    |` cres 4691   -->wf 5251   ` cfv 5255
This theorem is referenced by:  aomclem4  27154
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-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-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  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-fv 5263
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