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Theorem fnwe2lem3 26818
Description: Lemma for fnwe2 26819. 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 4514 . . . 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 5685 . . . . 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 5810 . . . 4  |-  ( ph  ->  ( ( F  |`  A ) `  a
)  e.  B )
96, 8eqeltrrd 2462 . . 3  |-  ( ph  ->  ( F `  a
)  e.  B )
10 fnwe2lem3.b . . . . 5  |-  ( ph  ->  b  e.  A )
11 fvres 5685 . . . . 5  |-  ( b  e.  A  ->  (
( F  |`  A ) `
 b )  =  ( F `  b
) )
1210, 11syl 16 . . . 4  |-  ( ph  ->  ( ( F  |`  A ) `  b
)  =  ( F `
 b ) )
137, 10ffvelrnd 5810 . . . 4  |-  ( ph  ->  ( ( F  |`  A ) `  b
)  e.  B )
1412, 13eqeltrrd 2462 . . 3  |-  ( ph  ->  ( F `  b
)  e.  B )
15 solin 4467 . . 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 1182 . 2  |-  ( ph  ->  ( ( F `  a ) R ( F `  b )  \/  ( F `  a )  =  ( F `  b )  \/  ( F `  b ) R ( F `  a ) ) )
17 orc 375 . . . . . 6  |-  ( ( F `  a ) R ( F `  b )  ->  (
( F `  a
) R ( F `
 b )  \/  ( ( F `  a )  =  ( F `  b )  /\  a [_ ( F `  a )  /  z ]_ S
b ) ) )
1817adantl 453 . . . . 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 26815 . . . . 5  |-  ( a T b  <->  ( ( F `  a ) R ( F `  b )  \/  (
( F `  a
)  =  ( F `
 b )  /\  a [_ ( F `  a )  /  z ]_ S b ) ) )
2218, 21sylibr 204 . . . 4  |-  ( (
ph  /\  ( F `  a ) R ( F `  b ) )  ->  a T
b )
23 3mix1 1126 . . . 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 26816 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  [_ ( F `  a )  /  z ]_ S  We  { y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
274, 26mpdan 650 . . . . . . 7  |-  ( ph  ->  [_ ( F `  a )  /  z ]_ S  We  { y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
28 weso 4514 . . . . . . 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 452 . . . . 5  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  [_ ( F `
 a )  / 
z ]_ S  Or  {
y  e.  A  | 
( F `  y
)  =  ( F `
 a ) } )
314adantr 452 . . . . . 6  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  a  e.  A )
32 eqidd 2388 . . . . . 6  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( F `  a )  =  ( F `  a ) )
33 fveq2 5668 . . . . . . . 8  |-  ( y  =  a  ->  ( F `  y )  =  ( F `  a ) )
3433eqeq1d 2395 . . . . . . 7  |-  ( y  =  a  ->  (
( F `  y
)  =  ( F `
 a )  <->  ( F `  a )  =  ( F `  a ) ) )
3534elrab 3035 . . . . . 6  |-  ( a  e.  { y  e.  A  |  ( F `
 y )  =  ( F `  a
) }  <->  ( a  e.  A  /\  ( F `  a )  =  ( F `  a ) ) )
3631, 32, 35sylanbrc 646 . . . . 5  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  a  e.  { y  e.  A  | 
( F `  y
)  =  ( F `
 a ) } )
3710adantr 452 . . . . . 6  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  b  e.  A )
38 simpr 448 . . . . . . 7  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( F `  a )  =  ( F `  b ) )
3938eqcomd 2392 . . . . . 6  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( F `  b )  =  ( F `  a ) )
40 fveq2 5668 . . . . . . . 8  |-  ( y  =  b  ->  ( F `  y )  =  ( F `  b ) )
4140eqeq1d 2395 . . . . . . 7  |-  ( y  =  b  ->  (
( F `  y
)  =  ( F `
 a )  <->  ( F `  b )  =  ( F `  a ) ) )
4241elrab 3035 . . . . . 6  |-  ( b  e.  { y  e.  A  |  ( F `
 y )  =  ( F `  a
) }  <->  ( b  e.  A  /\  ( F `  b )  =  ( F `  a ) ) )
4337, 39, 42sylanbrc 646 . . . . 5  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  b  e.  { y  e.  A  | 
( F `  y
)  =  ( F `
 a ) } )
44 solin 4467 . . . . 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 1182 . . . 4  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( a [_ ( F `  a
)  /  z ]_ S b  \/  a  =  b  \/  b [_ ( F `  a
)  /  z ]_ S a ) )
46 simplr 732 . . . . . . . . 9  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a [_ ( F `  a )  /  z ]_ S b )  -> 
( F `  a
)  =  ( F `
 b ) )
47 simpr 448 . . . . . . . . 9  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a [_ ( F `  a )  /  z ]_ S b )  -> 
a [_ ( F `  a )  /  z ]_ S b )
4846, 47jca 519 . . . . . . . 8  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a [_ ( F `  a )  /  z ]_ S b )  -> 
( ( F `  a )  =  ( F `  b )  /\  a [_ ( F `  a )  /  z ]_ S
b ) )
4948olcd 383 . . . . . . 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 204 . . . . . 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 1127 . . . . . 6  |-  ( a  =  b  ->  (
a T b  \/  a  =  b  \/  b T a ) )
5352adantl 453 . . . . 5  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a  =  b )  ->  ( a T b  \/  a  =  b  \/  b T a ) )
54 simplr 732 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
( F `  a
)  =  ( F `
 b ) )
5554eqcomd 2392 . . . . . . . . 9  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
( F `  b
)  =  ( F `
 a ) )
56 csbeq1 3197 . . . . . . . . . . . 12  |-  ( ( F `  a )  =  ( F `  b )  ->  [_ ( F `  a )  /  z ]_ S  =  [_ ( F `  b )  /  z ]_ S )
5756adantl 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  [_ ( F `
 a )  / 
z ]_ S  =  [_ ( F `  b )  /  z ]_ S
)
5857breqd 4164 . . . . . . . . . 10  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( b [_ ( F `  a
)  /  z ]_ S a  <->  b [_ ( F `  b )  /  z ]_ S
a ) )
5958biimpa 471 . . . . . . . . 9  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
b [_ ( F `  b )  /  z ]_ S a )
6055, 59jca 519 . . . . . . . 8  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
( ( F `  b )  =  ( F `  a )  /\  b [_ ( F `  b )  /  z ]_ S
a ) )
6160olcd 383 . . . . . . 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 26815 . . . . . . 7  |-  ( b T a  <->  ( ( F `  b ) R ( F `  a )  \/  (
( F `  b
)  =  ( F `
 a )  /\  b [_ ( F `  b )  /  z ]_ S a ) ) )
6361, 62sylibr 204 . . . . . 6  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
b T a )
64 3mix3 1128 . . . . . 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 1250 . . . 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 650 . . 3  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( a T b  \/  a  =  b  \/  b T a ) )
68 orc 375 . . . . . 6  |-  ( ( F `  b ) R ( F `  a )  ->  (
( F `  b
) R ( F `
 a )  \/  ( ( F `  b )  =  ( F `  a )  /\  b [_ ( F `  b )  /  z ]_ S
a ) ) )
6968adantl 453 . . . . 5  |-  ( (
ph  /\  ( F `  b ) R ( F `  a ) )  ->  ( ( F `  b ) R ( F `  a )  \/  (
( F `  b
)  =  ( F `
 a )  /\  b [_ ( F `  b )  /  z ]_ S a ) ) )
7069, 62sylibr 204 . . . 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 1250 . 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 650 1  |-  ( ph  ->  ( a T b  \/  a  =  b  \/  b T a ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    \/ w3o 935    = wceq 1649    e. wcel 1717   {crab 2653   [_csb 3194   class class class wbr 4153   {copab 4206    Or wor 4443    We wwe 4481    |` cres 4820   -->wf 5390   ` cfv 5394
This theorem is referenced by:  fnwe2  26819
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402
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