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Theorem fnwe2lem3 27141
Description: Lemma for fnwe2 27142. 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 4576 . . . 4  |-  ( R  We  B  ->  R  Or  B )
31, 2syl 16 . . 3  |-  ( ph  ->  R  Or  B )
4 fnwe2lem3.a . . . . 5  |-  ( ph  ->  a  e.  A )
5 fvres 5748 . . . . 5  |-  ( a  e.  A  ->  (
( F  |`  A ) `
 a )  =  ( F `  a
) )
64, 5syl 16 . . . 4  |-  ( ph  ->  ( ( F  |`  A ) `  a
)  =  ( F `
 a ) )
7 fnwe2.f . . . . 5  |-  ( ph  ->  ( F  |`  A ) : A --> B )
87, 4ffvelrnd 5874 . . . 4  |-  ( ph  ->  ( ( F  |`  A ) `  a
)  e.  B )
96, 8eqeltrrd 2513 . . 3  |-  ( ph  ->  ( F `  a
)  e.  B )
10 fnwe2lem3.b . . . . 5  |-  ( ph  ->  b  e.  A )
11 fvres 5748 . . . . 5  |-  ( b  e.  A  ->  (
( F  |`  A ) `
 b )  =  ( F `  b
) )
1210, 11syl 16 . . . 4  |-  ( ph  ->  ( ( F  |`  A ) `  b
)  =  ( F `
 b ) )
137, 10ffvelrnd 5874 . . . 4  |-  ( ph  ->  ( ( F  |`  A ) `  b
)  e.  B )
1412, 13eqeltrrd 2513 . . 3  |-  ( ph  ->  ( F `  b
)  e.  B )
15 solin 4529 . . 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 ) ) )
163, 9, 14, 15syl12anc 1183 . 2  |-  ( ph  ->  ( ( F `  a ) R ( F `  b )  \/  ( F `  a )  =  ( F `  b )  \/  ( F `  b ) R ( F `  a ) ) )
17 orc 376 . . . . . 6  |-  ( ( F `  a ) R ( F `  b )  ->  (
( F `  a
) R ( F `
 b )  \/  ( ( F `  a )  =  ( F `  b )  /\  a [_ ( F `  a )  /  z ]_ S
b ) ) )
1817adantl 454 . . . . 5  |-  ( (
ph  /\  ( F `  a ) R ( F `  b ) )  ->  ( ( F `  a ) R ( F `  b )  \/  (
( F `  a
)  =  ( F `
 b )  /\  a [_ ( F `  a )  /  z ]_ S b ) ) )
19 fnwe2.su . . . . . 6  |-  ( z  =  ( F `  x )  ->  S  =  U )
20 fnwe2.t . . . . . 6  |-  T  =  { <. x ,  y
>.  |  ( ( F `  x ) R ( F `  y )  \/  (
( F `  x
)  =  ( F `
 y )  /\  x U y ) ) }
2119, 20fnwe2val 27138 . . . . 5  |-  ( a T b  <->  ( ( F `  a ) R ( F `  b )  \/  (
( F `  a
)  =  ( F `
 b )  /\  a [_ ( F `  a )  /  z ]_ S b ) ) )
2218, 21sylibr 205 . . . 4  |-  ( (
ph  /\  ( F `  a ) R ( F `  b ) )  ->  a T
b )
23 3mix1 1127 . . . 4  |-  ( a T b  ->  (
a T b  \/  a  =  b  \/  b T a ) )
2422, 23syl 16 . . 3  |-  ( (
ph  /\  ( F `  a ) R ( F `  b ) )  ->  ( a T b  \/  a  =  b  \/  b T a ) )
25 fnwe2.s . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
2619, 20, 25fnwe2lem1 27139 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  [_ ( F `  a )  /  z ]_ S  We  { y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
274, 26mpdan 651 . . . . . . 7  |-  ( ph  ->  [_ ( F `  a )  /  z ]_ S  We  { y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
28 weso 4576 . . . . . . 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 ) } )
2927, 28syl 16 . . . . . 6  |-  ( ph  ->  [_ ( F `  a )  /  z ]_ S  Or  { y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
3029adantr 453 . . . . 5  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  [_ ( F `
 a )  / 
z ]_ S  Or  {
y  e.  A  | 
( F `  y
)  =  ( F `
 a ) } )
314adantr 453 . . . . . 6  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  a  e.  A )
32 eqidd 2439 . . . . . 6  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( F `  a )  =  ( F `  a ) )
33 fveq2 5731 . . . . . . . 8  |-  ( y  =  a  ->  ( F `  y )  =  ( F `  a ) )
3433eqeq1d 2446 . . . . . . 7  |-  ( y  =  a  ->  (
( F `  y
)  =  ( F `
 a )  <->  ( F `  a )  =  ( F `  a ) ) )
3534elrab 3094 . . . . . 6  |-  ( a  e.  { y  e.  A  |  ( F `
 y )  =  ( F `  a
) }  <->  ( a  e.  A  /\  ( F `  a )  =  ( F `  a ) ) )
3631, 32, 35sylanbrc 647 . . . . 5  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  a  e.  { y  e.  A  | 
( F `  y
)  =  ( F `
 a ) } )
3710adantr 453 . . . . . 6  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  b  e.  A )
38 simpr 449 . . . . . . 7  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( F `  a )  =  ( F `  b ) )
3938eqcomd 2443 . . . . . 6  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( F `  b )  =  ( F `  a ) )
40 fveq2 5731 . . . . . . . 8  |-  ( y  =  b  ->  ( F `  y )  =  ( F `  b ) )
4140eqeq1d 2446 . . . . . . 7  |-  ( y  =  b  ->  (
( F `  y
)  =  ( F `
 a )  <->  ( F `  b )  =  ( F `  a ) ) )
4241elrab 3094 . . . . . 6  |-  ( b  e.  { y  e.  A  |  ( F `
 y )  =  ( F `  a
) }  <->  ( b  e.  A  /\  ( F `  b )  =  ( F `  a ) ) )
4337, 39, 42sylanbrc 647 . . . . 5  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  b  e.  { y  e.  A  | 
( F `  y
)  =  ( F `
 a ) } )
44 solin 4529 . . . . 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 ) )
4530, 36, 43, 44syl12anc 1183 . . . 4  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( a [_ ( F `  a
)  /  z ]_ S b  \/  a  =  b  \/  b [_ ( F `  a
)  /  z ]_ S a ) )
46 simplr 733 . . . . . . . . 9  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a [_ ( F `  a )  /  z ]_ S b )  -> 
( F `  a
)  =  ( F `
 b ) )
47 simpr 449 . . . . . . . . 9  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a [_ ( F `  a )  /  z ]_ S b )  -> 
a [_ ( F `  a )  /  z ]_ S b )
4846, 47jca 520 . . . . . . . 8  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a [_ ( F `  a )  /  z ]_ S b )  -> 
( ( F `  a )  =  ( F `  b )  /\  a [_ ( F `  a )  /  z ]_ S
b ) )
4948olcd 384 . . . . . . 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 ) ) )
5049, 21sylibr 205 . . . . . 6  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a [_ ( F `  a )  /  z ]_ S b )  -> 
a T b )
5150, 23syl 16 . . . . 5  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a [_ ( F `  a )  /  z ]_ S b )  -> 
( a T b  \/  a  =  b  \/  b T a ) )
52 3mix2 1128 . . . . . 6  |-  ( a  =  b  ->  (
a T b  \/  a  =  b  \/  b T a ) )
5352adantl 454 . . . . 5  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a  =  b )  ->  ( a T b  \/  a  =  b  \/  b T a ) )
54 simplr 733 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
( F `  a
)  =  ( F `
 b ) )
5554eqcomd 2443 . . . . . . . . 9  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
( F `  b
)  =  ( F `
 a ) )
56 csbeq1 3256 . . . . . . . . . . . 12  |-  ( ( F `  a )  =  ( F `  b )  ->  [_ ( F `  a )  /  z ]_ S  =  [_ ( F `  b )  /  z ]_ S )
5756adantl 454 . . . . . . . . . . 11  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  [_ ( F `
 a )  / 
z ]_ S  =  [_ ( F `  b )  /  z ]_ S
)
5857breqd 4226 . . . . . . . . . 10  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( b [_ ( F `  a
)  /  z ]_ S a  <->  b [_ ( F `  b )  /  z ]_ S
a ) )
5958biimpa 472 . . . . . . . . 9  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
b [_ ( F `  b )  /  z ]_ S a )
6055, 59jca 520 . . . . . . . 8  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
( ( F `  b )  =  ( F `  a )  /\  b [_ ( F `  b )  /  z ]_ S
a ) )
6160olcd 384 . . . . . . 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 ) ) )
6219, 20fnwe2val 27138 . . . . . . 7  |-  ( b T a  <->  ( ( F `  b ) R ( F `  a )  \/  (
( F `  b
)  =  ( F `
 a )  /\  b [_ ( F `  b )  /  z ]_ S a ) ) )
6361, 62sylibr 205 . . . . . 6  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
b T a )
64 3mix3 1129 . . . . . 6  |-  ( b T a  ->  (
a T b  \/  a  =  b  \/  b T a ) )
6563, 64syl 16 . . . . 5  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
( a T b  \/  a  =  b  \/  b T a ) )
6651, 53, 653jaodan 1251 . . . 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 ) )
6745, 66mpdan 651 . . 3  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( a T b  \/  a  =  b  \/  b T a ) )
68 orc 376 . . . . . 6  |-  ( ( F `  b ) R ( F `  a )  ->  (
( F `  b
) R ( F `
 a )  \/  ( ( F `  b )  =  ( F `  a )  /\  b [_ ( F `  b )  /  z ]_ S
a ) ) )
6968adantl 454 . . . . 5  |-  ( (
ph  /\  ( F `  b ) R ( F `  a ) )  ->  ( ( F `  b ) R ( F `  a )  \/  (
( F `  b
)  =  ( F `
 a )  /\  b [_ ( F `  b )  /  z ]_ S a ) ) )
7069, 62sylibr 205 . . . 4  |-  ( (
ph  /\  ( F `  b ) R ( F `  a ) )  ->  b T
a )
7170, 64syl 16 . . 3  |-  ( (
ph  /\  ( F `  b ) R ( F `  a ) )  ->  ( a T b  \/  a  =  b  \/  b T a ) )
7224, 67, 713jaodan 1251 . 2  |-  ( (
ph  /\  ( ( F `  a ) R ( F `  b )  \/  ( F `  a )  =  ( F `  b )  \/  ( F `  b ) R ( F `  a ) ) )  ->  ( a T b  \/  a  =  b  \/  b T a ) )
7316, 72mpdan 651 1  |-  ( ph  ->  ( a T b  \/  a  =  b  \/  b T a ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    \/ w3o 936    = wceq 1653    e. wcel 1726   {crab 2711   [_csb 3253   class class class wbr 4215   {copab 4268    Or wor 4505    We wwe 4543    |` cres 4883   -->wf 5453   ` cfv 5457
This theorem is referenced by:  fnwe2  27142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465
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