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Theorem fnwe2 31238
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 6889 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 712 . . . . . 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 463 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  ( F  |`  A ) : A --> B )
7 fnwe2.r . . . . . . 7  |-  ( ph  ->  R  We  B )
87adantr 463 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  R  We  B )
9 simprl 754 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  a  C_  A )
10 simprr 755 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  a  =/=  (/) )
111, 2, 4, 6, 8, 9, 10fnwe2lem2 31236 . . . . 5  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  E. c  e.  a  A. d  e.  a  -.  d T c )
1211ex 432 . . . 4  |-  ( ph  ->  ( ( a  C_  A  /\  a  =/=  (/) )  ->  E. c  e.  a  A. d  e.  a  -.  d T c ) )
1312alrimiv 1724 . . 3  |-  ( ph  ->  A. a ( ( a  C_  A  /\  a  =/=  (/) )  ->  E. c  e.  a  A. d  e.  a  -.  d T c ) )
14 df-fr 4827 . . 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 712 . . . 4  |-  ( ( ( ph  /\  (
a  e.  A  /\  b  e.  A )
)  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
175adantr 463 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
( F  |`  A ) : A --> B )
187adantr 463 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  ->  R  We  B )
19 simprl 754 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
a  e.  A )
20 simprr 755 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
b  e.  A )
211, 2, 16, 17, 18, 19, 20fnwe2lem3 31237 . . 3  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
( a T b  \/  a  =  b  \/  b T a ) )
2221ralrimivva 2875 . 2  |-  ( ph  ->  A. a  e.  A  A. b  e.  A  ( a T b  \/  a  =  b  \/  b T a ) )
23 dfwe2 6590 . 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 662 1  |-  ( ph  ->  T  We  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 366    /\ wa 367    \/ w3o 970   A.wal 1396    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   {crab 2808    C_ wss 3461   (/)c0 3783   class class class wbr 4439   {copab 4496    Fr wfr 4824    We wwe 4826    |` cres 4990   -->wf 5566   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578
This theorem is referenced by:  aomclem4  31242
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