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Theorem fnwe2lem3 27149
Description: Lemma for fnwe2 27150. Trichotomy. (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 )
fnwe2lem3.a  |-  ( ph  ->  a  e.  A )
fnwe2lem3.b  |-  ( ph  ->  b  e.  A )
Assertion
Ref Expression
fnwe2lem3  |-  ( ph  ->  ( a T b  \/  a  =  b  \/  b T a ) )
Distinct variable groups:    y, U, z, a, b    x, S, y, a, b    x, R, y, a, b    ph, x, y, z    x, A, y, z, a, b    x, F, y, z, a, b    T, a, b    B, a, b
Allowed substitution hints:    ph( a, b)    B( x, y, z)    R( z)    S( z)    T( x, y, z)    U( x)

Proof of Theorem fnwe2lem3
StepHypRef Expression
1 fnwe2.r . . . 4  |-  ( ph  ->  R  We  B )
2 weso 4384 . . . 4  |-  ( R  We  B  ->  R  Or  B )
31, 2syl 15 . . 3  |-  ( ph  ->  R  Or  B )
4 fnwe2lem3.a . . . . 5  |-  ( ph  ->  a  e.  A )
5 fvres 5542 . . . . 5  |-  ( a  e.  A  ->  (
( F  |`  A ) `
 a )  =  ( F `  a
) )
64, 5syl 15 . . . 4  |-  ( ph  ->  ( ( F  |`  A ) `  a
)  =  ( F `
 a ) )
7 fnwe2.f . . . . 5  |-  ( ph  ->  ( F  |`  A ) : A --> B )
8 ffvelrn 5663 . . . . 5  |-  ( ( ( F  |`  A ) : A --> B  /\  a  e.  A )  ->  ( ( F  |`  A ) `  a
)  e.  B )
97, 4, 8syl2anc 642 . . . 4  |-  ( ph  ->  ( ( F  |`  A ) `  a
)  e.  B )
106, 9eqeltrrd 2358 . . 3  |-  ( ph  ->  ( F `  a
)  e.  B )
11 fnwe2lem3.b . . . . 5  |-  ( ph  ->  b  e.  A )
12 fvres 5542 . . . . 5  |-  ( b  e.  A  ->  (
( F  |`  A ) `
 b )  =  ( F `  b
) )
1311, 12syl 15 . . . 4  |-  ( ph  ->  ( ( F  |`  A ) `  b
)  =  ( F `
 b ) )
14 ffvelrn 5663 . . . . 5  |-  ( ( ( F  |`  A ) : A --> B  /\  b  e.  A )  ->  ( ( F  |`  A ) `  b
)  e.  B )
157, 11, 14syl2anc 642 . . . 4  |-  ( ph  ->  ( ( F  |`  A ) `  b
)  e.  B )
1613, 15eqeltrrd 2358 . . 3  |-  ( ph  ->  ( F `  b
)  e.  B )
17 solin 4337 . . 3  |-  ( ( R  Or  B  /\  ( ( F `  a )  e.  B  /\  ( F `  b
)  e.  B ) )  ->  ( ( F `  a ) R ( F `  b )  \/  ( F `  a )  =  ( F `  b )  \/  ( F `  b ) R ( F `  a ) ) )
183, 10, 16, 17syl12anc 1180 . 2  |-  ( ph  ->  ( ( F `  a ) R ( F `  b )  \/  ( F `  a )  =  ( F `  b )  \/  ( F `  b ) R ( F `  a ) ) )
19 orc 374 . . . . . 6  |-  ( ( F `  a ) R ( F `  b )  ->  (
( F `  a
) R ( F `
 b )  \/  ( ( F `  a )  =  ( F `  b )  /\  a [_ ( F `  a )  /  z ]_ S
b ) ) )
2019adantl 452 . . . . 5  |-  ( (
ph  /\  ( F `  a ) R ( F `  b ) )  ->  ( ( F `  a ) R ( F `  b )  \/  (
( F `  a
)  =  ( F `
 b )  /\  a [_ ( F `  a )  /  z ]_ S b ) ) )
21 fnwe2.su . . . . . 6  |-  ( z  =  ( F `  x )  ->  S  =  U )
22 fnwe2.t . . . . . 6  |-  T  =  { <. x ,  y
>.  |  ( ( F `  x ) R ( F `  y )  \/  (
( F `  x
)  =  ( F `
 y )  /\  x U y ) ) }
2321, 22fnwe2val 27146 . . . . 5  |-  ( a T b  <->  ( ( F `  a ) R ( F `  b )  \/  (
( F `  a
)  =  ( F `
 b )  /\  a [_ ( F `  a )  /  z ]_ S b ) ) )
2420, 23sylibr 203 . . . 4  |-  ( (
ph  /\  ( F `  a ) R ( F `  b ) )  ->  a T
b )
25 3mix1 1124 . . . 4  |-  ( a T b  ->  (
a T b  \/  a  =  b  \/  b T a ) )
2624, 25syl 15 . . 3  |-  ( (
ph  /\  ( F `  a ) R ( F `  b ) )  ->  ( a T b  \/  a  =  b  \/  b T a ) )
27 fnwe2.s . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
2821, 22, 27fnwe2lem1 27147 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  [_ ( F `  a )  /  z ]_ S  We  { y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
294, 28mpdan 649 . . . . . . 7  |-  ( ph  ->  [_ ( F `  a )  /  z ]_ S  We  { y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
30 weso 4384 . . . . . . 7  |-  ( [_ ( F `  a )  /  z ]_ S  We  { y  e.  A  |  ( F `  y )  =  ( F `  a ) }  ->  [_ ( F `
 a )  / 
z ]_ S  Or  {
y  e.  A  | 
( F `  y
)  =  ( F `
 a ) } )
3129, 30syl 15 . . . . . 6  |-  ( ph  ->  [_ ( F `  a )  /  z ]_ S  Or  { y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
3231adantr 451 . . . . 5  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  [_ ( F `
 a )  / 
z ]_ S  Or  {
y  e.  A  | 
( F `  y
)  =  ( F `
 a ) } )
334adantr 451 . . . . . 6  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  a  e.  A )
34 eqidd 2284 . . . . . 6  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( F `  a )  =  ( F `  a ) )
35 fveq2 5525 . . . . . . . 8  |-  ( y  =  a  ->  ( F `  y )  =  ( F `  a ) )
3635eqeq1d 2291 . . . . . . 7  |-  ( y  =  a  ->  (
( F `  y
)  =  ( F `
 a )  <->  ( F `  a )  =  ( F `  a ) ) )
3736elrab 2923 . . . . . 6  |-  ( a  e.  { y  e.  A  |  ( F `
 y )  =  ( F `  a
) }  <->  ( a  e.  A  /\  ( F `  a )  =  ( F `  a ) ) )
3833, 34, 37sylanbrc 645 . . . . 5  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  a  e.  { y  e.  A  | 
( F `  y
)  =  ( F `
 a ) } )
3911adantr 451 . . . . . 6  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  b  e.  A )
40 simpr 447 . . . . . . 7  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( F `  a )  =  ( F `  b ) )
4140eqcomd 2288 . . . . . 6  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( F `  b )  =  ( F `  a ) )
42 fveq2 5525 . . . . . . . 8  |-  ( y  =  b  ->  ( F `  y )  =  ( F `  b ) )
4342eqeq1d 2291 . . . . . . 7  |-  ( y  =  b  ->  (
( F `  y
)  =  ( F `
 a )  <->  ( F `  b )  =  ( F `  a ) ) )
4443elrab 2923 . . . . . 6  |-  ( b  e.  { y  e.  A  |  ( F `
 y )  =  ( F `  a
) }  <->  ( b  e.  A  /\  ( F `  b )  =  ( F `  a ) ) )
4539, 41, 44sylanbrc 645 . . . . 5  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  b  e.  { y  e.  A  | 
( F `  y
)  =  ( F `
 a ) } )
46 solin 4337 . . . . 5  |-  ( (
[_ ( F `  a )  /  z ]_ S  Or  { y  e.  A  |  ( F `  y )  =  ( F `  a ) }  /\  ( a  e.  {
y  e.  A  | 
( F `  y
)  =  ( F `
 a ) }  /\  b  e.  {
y  e.  A  | 
( F `  y
)  =  ( F `
 a ) } ) )  ->  (
a [_ ( F `  a )  /  z ]_ S b  \/  a  =  b  \/  b [_ ( F `  a
)  /  z ]_ S a ) )
4732, 38, 45, 46syl12anc 1180 . . . 4  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( a [_ ( F `  a
)  /  z ]_ S b  \/  a  =  b  \/  b [_ ( F `  a
)  /  z ]_ S a ) )
48 simplr 731 . . . . . . . . 9  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a [_ ( F `  a )  /  z ]_ S b )  -> 
( F `  a
)  =  ( F `
 b ) )
49 simpr 447 . . . . . . . . 9  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a [_ ( F `  a )  /  z ]_ S b )  -> 
a [_ ( F `  a )  /  z ]_ S b )
5048, 49jca 518 . . . . . . . 8  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a [_ ( F `  a )  /  z ]_ S b )  -> 
( ( F `  a )  =  ( F `  b )  /\  a [_ ( F `  a )  /  z ]_ S
b ) )
5150olcd 382 . . . . . . 7  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a [_ ( F `  a )  /  z ]_ S b )  -> 
( ( F `  a ) R ( F `  b )  \/  ( ( F `
 a )  =  ( F `  b
)  /\  a [_ ( F `  a )  /  z ]_ S
b ) ) )
5251, 23sylibr 203 . . . . . 6  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a [_ ( F `  a )  /  z ]_ S b )  -> 
a T b )
5352, 25syl 15 . . . . 5  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a [_ ( F `  a )  /  z ]_ S b )  -> 
( a T b  \/  a  =  b  \/  b T a ) )
54 3mix2 1125 . . . . . 6  |-  ( a  =  b  ->  (
a T b  \/  a  =  b  \/  b T a ) )
5554adantl 452 . . . . 5  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a  =  b )  ->  ( a T b  \/  a  =  b  \/  b T a ) )
56 simplr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
( F `  a
)  =  ( F `
 b ) )
5756eqcomd 2288 . . . . . . . . 9  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
( F `  b
)  =  ( F `
 a ) )
58 csbeq1 3084 . . . . . . . . . . . 12  |-  ( ( F `  a )  =  ( F `  b )  ->  [_ ( F `  a )  /  z ]_ S  =  [_ ( F `  b )  /  z ]_ S )
5958adantl 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  [_ ( F `
 a )  / 
z ]_ S  =  [_ ( F `  b )  /  z ]_ S
)
6059breqd 4034 . . . . . . . . . 10  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( b [_ ( F `  a
)  /  z ]_ S a  <->  b [_ ( F `  b )  /  z ]_ S
a ) )
6160biimpa 470 . . . . . . . . 9  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
b [_ ( F `  b )  /  z ]_ S a )
6257, 61jca 518 . . . . . . . 8  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
( ( F `  b )  =  ( F `  a )  /\  b [_ ( F `  b )  /  z ]_ S
a ) )
6362olcd 382 . . . . . . 7  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
( ( F `  b ) R ( F `  a )  \/  ( ( F `
 b )  =  ( F `  a
)  /\  b [_ ( F `  b )  /  z ]_ S
a ) ) )
6421, 22fnwe2val 27146 . . . . . . 7  |-  ( b T a  <->  ( ( F `  b ) R ( F `  a )  \/  (
( F `  b
)  =  ( F `
 a )  /\  b [_ ( F `  b )  /  z ]_ S a ) ) )
6563, 64sylibr 203 . . . . . 6  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
b T a )
66 3mix3 1126 . . . . . 6  |-  ( b T a  ->  (
a T b  \/  a  =  b  \/  b T a ) )
6765, 66syl 15 . . . . 5  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
( a T b  \/  a  =  b  \/  b T a ) )
6853, 55, 673jaodan 1248 . . . 4  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  ( a [_ ( F `  a )  /  z ]_ S
b  \/  a  =  b  \/  b [_ ( F `  a )  /  z ]_ S
a ) )  -> 
( a T b  \/  a  =  b  \/  b T a ) )
6947, 68mpdan 649 . . 3  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( a T b  \/  a  =  b  \/  b T a ) )
70 orc 374 . . . . . 6  |-  ( ( F `  b ) R ( F `  a )  ->  (
( F `  b
) R ( F `
 a )  \/  ( ( F `  b )  =  ( F `  a )  /\  b [_ ( F `  b )  /  z ]_ S
a ) ) )
7170adantl 452 . . . . 5  |-  ( (
ph  /\  ( F `  b ) R ( F `  a ) )  ->  ( ( F `  b ) R ( F `  a )  \/  (
( F `  b
)  =  ( F `
 a )  /\  b [_ ( F `  b )  /  z ]_ S a ) ) )
7271, 64sylibr 203 . . . 4  |-  ( (
ph  /\  ( F `  b ) R ( F `  a ) )  ->  b T
a )
7372, 66syl 15 . . 3  |-  ( (
ph  /\  ( F `  b ) R ( F `  a ) )  ->  ( a T b  \/  a  =  b  \/  b T a ) )
7426, 69, 733jaodan 1248 . 2  |-  ( (
ph  /\  ( ( F `  a ) R ( F `  b )  \/  ( F `  a )  =  ( F `  b )  \/  ( F `  b ) R ( F `  a ) ) )  ->  ( a T b  \/  a  =  b  \/  b T a ) )
7518, 74mpdan 649 1  |-  ( ph  ->  ( a T b  \/  a  =  b  \/  b T a ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    \/ w3o 933    = wceq 1623    e. wcel 1684   {crab 2547   [_csb 3081   class class class wbr 4023   {copab 4076    Or wor 4313    We wwe 4351    |` cres 4691   -->wf 5251   ` cfv 5255
This theorem is referenced by:  fnwe2  27150
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-op 3649  df-uni 3828  df-br 4024  df-opab 4078  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263
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