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Theorem fnwe2 29553
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 6797 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 714 . . . . . 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 465 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  ( F  |`  A ) : A --> B )
7 fnwe2.r . . . . . . 7  |-  ( ph  ->  R  We  B )
87adantr 465 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  R  We  B )
9 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  a  C_  A )
10 simprr 756 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  a  =/=  (/) )
111, 2, 4, 6, 8, 9, 10fnwe2lem2 29551 . . . . 5  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  E. c  e.  a  A. d  e.  a  -.  d T c )
1211ex 434 . . . 4  |-  ( ph  ->  ( ( a  C_  A  /\  a  =/=  (/) )  ->  E. c  e.  a  A. d  e.  a  -.  d T c ) )
1312alrimiv 1686 . . 3  |-  ( ph  ->  A. a ( ( a  C_  A  /\  a  =/=  (/) )  ->  E. c  e.  a  A. d  e.  a  -.  d T c ) )
14 df-fr 4786 . . 3  |-  ( T  Fr  A  <->  A. a
( ( a  C_  A  /\  a  =/=  (/) )  ->  E. c  e.  a  A. d  e.  a  -.  d T c ) )
1513, 14sylibr 212 . 2  |-  ( ph  ->  T  Fr  A )
163adantlr 714 . . . 4  |-  ( ( ( ph  /\  (
a  e.  A  /\  b  e.  A )
)  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
175adantr 465 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
( F  |`  A ) : A --> B )
187adantr 465 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  ->  R  We  B )
19 simprl 755 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
a  e.  A )
20 simprr 756 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
b  e.  A )
211, 2, 16, 17, 18, 19, 20fnwe2lem3 29552 . . 3  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
( a T b  \/  a  =  b  \/  b T a ) )
2221ralrimivva 2912 . 2  |-  ( ph  ->  A. a  e.  A  A. b  e.  A  ( a T b  \/  a  =  b  \/  b T a ) )
23 dfwe2 6502 . 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 664 1  |-  ( ph  ->  T  We  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    \/ w3o 964   A.wal 1368    = wceq 1370    e. wcel 1758    =/= wne 2647   A.wral 2798   E.wrex 2799   {crab 2802    C_ wss 3435   (/)c0 3744   class class class wbr 4399   {copab 4456    Fr wfr 4783    We wwe 4785    |` cres 4949   -->wf 5521   ` cfv 5525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533
This theorem is referenced by:  aomclem4  29557
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