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Theorem fnwe2 27119
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 6454 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 696 . . . . . 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 452 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  ( F  |`  A ) : A --> B )
7 fnwe2.r . . . . . . 7  |-  ( ph  ->  R  We  B )
87adantr 452 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  R  We  B )
9 simprl 733 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  a  C_  A )
10 simprr 734 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  a  =/=  (/) )
111, 2, 4, 6, 8, 9, 10fnwe2lem2 27117 . . . . 5  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  E. c  e.  a  A. d  e.  a  -.  d T c )
1211ex 424 . . . 4  |-  ( ph  ->  ( ( a  C_  A  /\  a  =/=  (/) )  ->  E. c  e.  a  A. d  e.  a  -.  d T c ) )
1312alrimiv 1641 . . 3  |-  ( ph  ->  A. a ( ( a  C_  A  /\  a  =/=  (/) )  ->  E. c  e.  a  A. d  e.  a  -.  d T c ) )
14 df-fr 4533 . . 3  |-  ( T  Fr  A  <->  A. a
( ( a  C_  A  /\  a  =/=  (/) )  ->  E. c  e.  a  A. d  e.  a  -.  d T c ) )
1513, 14sylibr 204 . 2  |-  ( ph  ->  T  Fr  A )
163adantlr 696 . . . 4  |-  ( ( ( ph  /\  (
a  e.  A  /\  b  e.  A )
)  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
175adantr 452 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
( F  |`  A ) : A --> B )
187adantr 452 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  ->  R  We  B )
19 simprl 733 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
a  e.  A )
20 simprr 734 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
b  e.  A )
211, 2, 16, 17, 18, 19, 20fnwe2lem3 27118 . . 3  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
( a T b  \/  a  =  b  \/  b T a ) )
2221ralrimivva 2790 . 2  |-  ( ph  ->  A. a  e.  A  A. b  e.  A  ( a T b  \/  a  =  b  \/  b T a ) )
23 dfwe2 4754 . 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 646 1  |-  ( ph  ->  T  We  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    \/ w3o 935   A.wal 1549    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   {crab 2701    C_ wss 3312   (/)c0 3620   class class class wbr 4204   {copab 4257    Fr wfr 4530    We wwe 4532    |` cres 4872   -->wf 5442   ` cfv 5446
This theorem is referenced by:  aomclem4  27123
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-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-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  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-fv 5454
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