Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fnwe2 Structured version   Visualization version   Unicode version

Theorem fnwe2 35923
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 6917 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 722 . . . . . 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 467 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  ( F  |`  A ) : A --> B )
7 fnwe2.r . . . . . . 7  |-  ( ph  ->  R  We  B )
87adantr 467 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  R  We  B )
9 simprl 765 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  a  C_  A )
10 simprr 767 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  a  =/=  (/) )
111, 2, 4, 6, 8, 9, 10fnwe2lem2 35921 . . . . 5  |-  ( (
ph  /\  ( a  C_  A  /\  a  =/=  (/) ) )  ->  E. c  e.  a  A. d  e.  a  -.  d T c )
1211ex 436 . . . 4  |-  ( ph  ->  ( ( a  C_  A  /\  a  =/=  (/) )  ->  E. c  e.  a  A. d  e.  a  -.  d T c ) )
1312alrimiv 1775 . . 3  |-  ( ph  ->  A. a ( ( a  C_  A  /\  a  =/=  (/) )  ->  E. c  e.  a  A. d  e.  a  -.  d T c ) )
14 df-fr 4796 . . 3  |-  ( T  Fr  A  <->  A. a
( ( a  C_  A  /\  a  =/=  (/) )  ->  E. c  e.  a  A. d  e.  a  -.  d T c ) )
1513, 14sylibr 216 . 2  |-  ( ph  ->  T  Fr  A )
163adantlr 722 . . . 4  |-  ( ( ( ph  /\  (
a  e.  A  /\  b  e.  A )
)  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
175adantr 467 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
( F  |`  A ) : A --> B )
187adantr 467 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  ->  R  We  B )
19 simprl 765 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
a  e.  A )
20 simprr 767 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
b  e.  A )
211, 2, 16, 17, 18, 19, 20fnwe2lem3 35922 . . 3  |-  ( (
ph  /\  ( a  e.  A  /\  b  e.  A ) )  -> 
( a T b  \/  a  =  b  \/  b T a ) )
2221ralrimivva 2811 . 2  |-  ( ph  ->  A. a  e.  A  A. b  e.  A  ( a T b  \/  a  =  b  \/  b T a ) )
23 dfwe2 6613 . 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 671 1  |-  ( ph  ->  T  We  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 370    /\ wa 371    \/ w3o 985   A.wal 1444    = wceq 1446    e. wcel 1889    =/= wne 2624   A.wral 2739   E.wrex 2740   {crab 2743    C_ wss 3406   (/)c0 3733   class class class wbr 4405   {copab 4463    Fr wfr 4793    We wwe 4795    |` cres 4839   -->wf 5581   ` cfv 5585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-fv 5593
This theorem is referenced by:  aomclem4  35927
  Copyright terms: Public domain W3C validator