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Theorem fnwe2lem3 35340
Description: Lemma for fnwe2 35341. 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 orc 383 . . . . 5  |-  ( ( F `  a ) R ( F `  b )  ->  (
( F `  a
) R ( F `
 b )  \/  ( ( F `  a )  =  ( F `  b )  /\  a [_ ( F `  a )  /  z ]_ S
b ) ) )
21adantl 464 . . . 4  |-  ( (
ph  /\  ( F `  a ) R ( F `  b ) )  ->  ( ( F `  a ) R ( F `  b )  \/  (
( F `  a
)  =  ( F `
 b )  /\  a [_ ( F `  a )  /  z ]_ S b ) ) )
3 fnwe2.su . . . . 5  |-  ( z  =  ( F `  x )  ->  S  =  U )
4 fnwe2.t . . . . 5  |-  T  =  { <. x ,  y
>.  |  ( ( F `  x ) R ( F `  y )  \/  (
( F `  x
)  =  ( F `
 y )  /\  x U y ) ) }
53, 4fnwe2val 35337 . . . 4  |-  ( a T b  <->  ( ( F `  a ) R ( F `  b )  \/  (
( F `  a
)  =  ( F `
 b )  /\  a [_ ( F `  a )  /  z ]_ S b ) ) )
62, 5sylibr 212 . . 3  |-  ( (
ph  /\  ( F `  a ) R ( F `  b ) )  ->  a T
b )
763mix1d 1172 . 2  |-  ( (
ph  /\  ( F `  a ) R ( F `  b ) )  ->  ( a T b  \/  a  =  b  \/  b T a ) )
8 simplr 754 . . . . . . 7  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a [_ ( F `  a )  /  z ]_ S b )  -> 
( F `  a
)  =  ( F `
 b ) )
9 simpr 459 . . . . . . 7  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a [_ ( F `  a )  /  z ]_ S b )  -> 
a [_ ( F `  a )  /  z ]_ S b )
108, 9jca 530 . . . . . 6  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a [_ ( F `  a )  /  z ]_ S b )  -> 
( ( F `  a )  =  ( F `  b )  /\  a [_ ( F `  a )  /  z ]_ S
b ) )
1110olcd 391 . . . . 5  |-  ( ( ( 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 ) ) )
1211, 5sylibr 212 . . . 4  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a [_ ( F `  a )  /  z ]_ S b )  -> 
a T b )
13123mix1d 1172 . . 3  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a [_ ( F `  a )  /  z ]_ S b )  -> 
( a T b  \/  a  =  b  \/  b T a ) )
14 3mix2 1167 . . . 4  |-  ( a  =  b  ->  (
a T b  \/  a  =  b  \/  b T a ) )
1514adantl 464 . . 3  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  a  =  b )  ->  ( a T b  \/  a  =  b  \/  b T a ) )
16 simplr 754 . . . . . . . 8  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
( F `  a
)  =  ( F `
 b ) )
1716eqcomd 2410 . . . . . . 7  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
( F `  b
)  =  ( F `
 a ) )
18 csbeq1 3375 . . . . . . . . . 10  |-  ( ( F `  a )  =  ( F `  b )  ->  [_ ( F `  a )  /  z ]_ S  =  [_ ( F `  b )  /  z ]_ S )
1918adantl 464 . . . . . . . . 9  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  [_ ( F `
 a )  / 
z ]_ S  =  [_ ( F `  b )  /  z ]_ S
)
2019breqd 4405 . . . . . . . 8  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( b [_ ( F `  a
)  /  z ]_ S a  <->  b [_ ( F `  b )  /  z ]_ S
a ) )
2120biimpa 482 . . . . . . 7  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
b [_ ( F `  b )  /  z ]_ S a )
2217, 21jca 530 . . . . . 6  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
( ( F `  b )  =  ( F `  a )  /\  b [_ ( F `  b )  /  z ]_ S
a ) )
2322olcd 391 . . . . 5  |-  ( ( ( 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 ) ) )
243, 4fnwe2val 35337 . . . . 5  |-  ( b T a  <->  ( ( F `  b ) R ( F `  a )  \/  (
( F `  b
)  =  ( F `
 a )  /\  b [_ ( F `  b )  /  z ]_ S a ) ) )
2523, 24sylibr 212 . . . 4  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
b T a )
26253mix3d 1174 . . 3  |-  ( ( ( ph  /\  ( F `  a )  =  ( F `  b ) )  /\  b [_ ( F `  a )  /  z ]_ S a )  -> 
( a T b  \/  a  =  b  \/  b T a ) )
27 fnwe2lem3.a . . . . . . 7  |-  ( ph  ->  a  e.  A )
28 fnwe2.s . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
293, 4, 28fnwe2lem1 35338 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  [_ ( F `  a )  /  z ]_ S  We  { y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
3027, 29mpdan 666 . . . . . 6  |-  ( ph  ->  [_ ( F `  a )  /  z ]_ S  We  { y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
31 weso 4813 . . . . . 6  |-  ( [_ ( F `  a )  /  z ]_ S  We  { y  e.  A  |  ( F `  y )  =  ( F `  a ) }  ->  [_ ( F `
 a )  / 
z ]_ S  Or  {
y  e.  A  | 
( F `  y
)  =  ( F `
 a ) } )
3230, 31syl 17 . . . . 5  |-  ( ph  ->  [_ ( F `  a )  /  z ]_ S  Or  { y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
3332adantr 463 . . . 4  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  [_ ( F `
 a )  / 
z ]_ S  Or  {
y  e.  A  | 
( F `  y
)  =  ( F `
 a ) } )
3427adantr 463 . . . . 5  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  a  e.  A )
35 eqidd 2403 . . . . 5  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( F `  a )  =  ( F `  a ) )
36 fveq2 5848 . . . . . . 7  |-  ( y  =  a  ->  ( F `  y )  =  ( F `  a ) )
3736eqeq1d 2404 . . . . . 6  |-  ( y  =  a  ->  (
( F `  y
)  =  ( F `
 a )  <->  ( F `  a )  =  ( F `  a ) ) )
3837elrab 3206 . . . . 5  |-  ( a  e.  { y  e.  A  |  ( F `
 y )  =  ( F `  a
) }  <->  ( a  e.  A  /\  ( F `  a )  =  ( F `  a ) ) )
3934, 35, 38sylanbrc 662 . . . 4  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  a  e.  { y  e.  A  | 
( F `  y
)  =  ( F `
 a ) } )
40 fnwe2lem3.b . . . . . 6  |-  ( ph  ->  b  e.  A )
4140adantr 463 . . . . 5  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  b  e.  A )
42 simpr 459 . . . . . 6  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( F `  a )  =  ( F `  b ) )
4342eqcomd 2410 . . . . 5  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( F `  b )  =  ( F `  a ) )
44 fveq2 5848 . . . . . . 7  |-  ( y  =  b  ->  ( F `  y )  =  ( F `  b ) )
4544eqeq1d 2404 . . . . . 6  |-  ( y  =  b  ->  (
( F `  y
)  =  ( F `
 a )  <->  ( F `  b )  =  ( F `  a ) ) )
4645elrab 3206 . . . . 5  |-  ( b  e.  { y  e.  A  |  ( F `
 y )  =  ( F `  a
) }  <->  ( b  e.  A  /\  ( F `  b )  =  ( F `  a ) ) )
4741, 43, 46sylanbrc 662 . . . 4  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  b  e.  { y  e.  A  | 
( F `  y
)  =  ( F `
 a ) } )
48 solin 4766 . . . 4  |-  ( (
[_ ( 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 ) )
4933, 39, 47, 48syl12anc 1228 . . 3  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( a [_ ( F `  a
)  /  z ]_ S b  \/  a  =  b  \/  b [_ ( F `  a
)  /  z ]_ S a ) )
5013, 15, 26, 49mpjao3dan 1297 . 2  |-  ( (
ph  /\  ( F `  a )  =  ( F `  b ) )  ->  ( a T b  \/  a  =  b  \/  b T a ) )
51 orc 383 . . . . 5  |-  ( ( F `  b ) R ( F `  a )  ->  (
( F `  b
) R ( F `
 a )  \/  ( ( F `  b )  =  ( F `  a )  /\  b [_ ( F `  b )  /  z ]_ S
a ) ) )
5251adantl 464 . . . 4  |-  ( (
ph  /\  ( F `  b ) R ( F `  a ) )  ->  ( ( F `  b ) R ( F `  a )  \/  (
( F `  b
)  =  ( F `
 a )  /\  b [_ ( F `  b )  /  z ]_ S a ) ) )
5352, 24sylibr 212 . . 3  |-  ( (
ph  /\  ( F `  b ) R ( F `  a ) )  ->  b T
a )
54533mix3d 1174 . 2  |-  ( (
ph  /\  ( F `  b ) R ( F `  a ) )  ->  ( a T b  \/  a  =  b  \/  b T a ) )
55 fnwe2.r . . . 4  |-  ( ph  ->  R  We  B )
56 weso 4813 . . . 4  |-  ( R  We  B  ->  R  Or  B )
5755, 56syl 17 . . 3  |-  ( ph  ->  R  Or  B )
58 fvres 5862 . . . . 5  |-  ( a  e.  A  ->  (
( F  |`  A ) `
 a )  =  ( F `  a
) )
5927, 58syl 17 . . . 4  |-  ( ph  ->  ( ( F  |`  A ) `  a
)  =  ( F `
 a ) )
60 fnwe2.f . . . . 5  |-  ( ph  ->  ( F  |`  A ) : A --> B )
6160, 27ffvelrnd 6009 . . . 4  |-  ( ph  ->  ( ( F  |`  A ) `  a
)  e.  B )
6259, 61eqeltrrd 2491 . . 3  |-  ( ph  ->  ( F `  a
)  e.  B )
63 fvres 5862 . . . . 5  |-  ( b  e.  A  ->  (
( F  |`  A ) `
 b )  =  ( F `  b
) )
6440, 63syl 17 . . . 4  |-  ( ph  ->  ( ( F  |`  A ) `  b
)  =  ( F `
 b ) )
6560, 40ffvelrnd 6009 . . . 4  |-  ( ph  ->  ( ( F  |`  A ) `  b
)  e.  B )
6664, 65eqeltrrd 2491 . . 3  |-  ( ph  ->  ( F `  b
)  e.  B )
67 solin 4766 . . 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 ) ) )
6857, 62, 66, 67syl12anc 1228 . 2  |-  ( ph  ->  ( ( F `  a ) R ( F `  b )  \/  ( F `  a )  =  ( F `  b )  \/  ( F `  b ) R ( F `  a ) ) )
697, 50, 54, 68mpjao3dan 1297 1  |-  ( ph  ->  ( a T b  \/  a  =  b  \/  b T a ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    \/ w3o 973    = wceq 1405    e. wcel 1842   {crab 2757   [_csb 3372   class class class wbr 4394   {copab 4451    Or wor 4742    We wwe 4780    |` cres 4824   -->wf 5564   ` cfv 5568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fv 5576
This theorem is referenced by:  fnwe2  35341
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