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Theorem wemapso 7765
Description: Construct lexicographic order on a function space based on a well-ordering of the indexes and a total ordering of the values. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-2015.)
Hypothesis
Ref Expression
wemapso.t  |-  T  =  { <. x ,  y
>.  |  E. z  e.  A  ( (
x `  z ) S ( y `  z )  /\  A. w  e.  A  (
w R z  -> 
( x `  w
)  =  ( y `
 w ) ) ) }
Assertion
Ref Expression
wemapso  |-  ( ( A  e.  V  /\  R  We  A  /\  S  Or  B )  ->  T  Or  ( B  ^m  A ) )
Distinct variable groups:    x, B    x, w, y, z, A   
w, R, x, y, z    w, S, x, y, z
Allowed substitution hints:    B( y, z, w)    T( x, y, z, w)    V( x, y, z, w)

Proof of Theorem wemapso
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2981 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 wemapso.t . . 3  |-  T  =  { <. x ,  y
>.  |  E. z  e.  A  ( (
x `  z ) S ( y `  z )  /\  A. w  e.  A  (
w R z  -> 
( x `  w
)  =  ( y `
 w ) ) ) }
3 ssid 3375 . . 3  |-  ( B  ^m  A )  C_  ( B  ^m  A )
4 simp1 988 . . 3  |-  ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  ->  A  e.  _V )
5 weso 4711 . . . 4  |-  ( R  We  A  ->  R  Or  A )
653ad2ant2 1010 . . 3  |-  ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  ->  R  Or  A )
7 simp3 990 . . 3  |-  ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  ->  S  Or  B )
8 simpl1 991 . . . . 5  |-  ( ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  /\  ( ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A ) )  /\  a  =/=  b ) )  ->  A  e.  _V )
9 difss 3483 . . . . . . 7  |-  ( a 
\  b )  C_  a
10 dmss 5039 . . . . . . 7  |-  ( ( a  \  b ) 
C_  a  ->  dom  ( a  \  b
)  C_  dom  a )
119, 10ax-mp 5 . . . . . 6  |-  dom  (
a  \  b )  C_ 
dom  a
12 simprll 761 . . . . . . . . 9  |-  ( ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  /\  ( ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A ) )  /\  a  =/=  b ) )  -> 
a  e.  ( B  ^m  A ) )
13 elmapi 7234 . . . . . . . . 9  |-  ( a  e.  ( B  ^m  A )  ->  a : A --> B )
1412, 13syl 16 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  /\  ( ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A ) )  /\  a  =/=  b ) )  -> 
a : A --> B )
15 ffn 5559 . . . . . . . 8  |-  ( a : A --> B  -> 
a  Fn  A )
1614, 15syl 16 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  /\  ( ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A ) )  /\  a  =/=  b ) )  -> 
a  Fn  A )
17 fndm 5510 . . . . . . 7  |-  ( a  Fn  A  ->  dom  a  =  A )
1816, 17syl 16 . . . . . 6  |-  ( ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  /\  ( ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A ) )  /\  a  =/=  b ) )  ->  dom  a  =  A
)
1911, 18syl5sseq 3404 . . . . 5  |-  ( ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  /\  ( ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A ) )  /\  a  =/=  b ) )  ->  dom  ( a  \  b
)  C_  A )
208, 19ssexd 4439 . . . 4  |-  ( ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  /\  ( ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A ) )  /\  a  =/=  b ) )  ->  dom  ( a  \  b
)  e.  _V )
21 simpl2 992 . . . . 5  |-  ( ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  /\  ( ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A ) )  /\  a  =/=  b ) )  ->  R  We  A )
22 wefr 4710 . . . . 5  |-  ( R  We  A  ->  R  Fr  A )
2321, 22syl 16 . . . 4  |-  ( ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  /\  ( ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A ) )  /\  a  =/=  b ) )  ->  R  Fr  A )
24 simprr 756 . . . . 5  |-  ( ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  /\  ( ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A ) )  /\  a  =/=  b ) )  -> 
a  =/=  b )
25 simprlr 762 . . . . . . . . 9  |-  ( ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  /\  ( ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A ) )  /\  a  =/=  b ) )  -> 
b  e.  ( B  ^m  A ) )
26 elmapi 7234 . . . . . . . . 9  |-  ( b  e.  ( B  ^m  A )  ->  b : A --> B )
2725, 26syl 16 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  /\  ( ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A ) )  /\  a  =/=  b ) )  -> 
b : A --> B )
28 ffn 5559 . . . . . . . 8  |-  ( b : A --> B  -> 
b  Fn  A )
2927, 28syl 16 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  /\  ( ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A ) )  /\  a  =/=  b ) )  -> 
b  Fn  A )
30 fndmdifeq0 5809 . . . . . . 7  |-  ( ( a  Fn  A  /\  b  Fn  A )  ->  ( dom  ( a 
\  b )  =  (/) 
<->  a  =  b ) )
3116, 29, 30syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  /\  ( ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A ) )  /\  a  =/=  b ) )  -> 
( dom  ( a  \  b )  =  (/) 
<->  a  =  b ) )
3231necon3bid 2643 . . . . 5  |-  ( ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  /\  ( ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A ) )  /\  a  =/=  b ) )  -> 
( dom  ( a  \  b )  =/=  (/) 
<->  a  =/=  b ) )
3324, 32mpbird 232 . . . 4  |-  ( ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  /\  ( ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A ) )  /\  a  =/=  b ) )  ->  dom  ( a  \  b
)  =/=  (/) )
34 fri 4682 . . . 4  |-  ( ( ( dom  ( a 
\  b )  e. 
_V  /\  R  Fr  A )  /\  ( dom  ( a  \  b
)  C_  A  /\  dom  ( a  \  b
)  =/=  (/) ) )  ->  E. c  e.  dom  ( a  \  b
) A. d  e. 
dom  ( a  \ 
b )  -.  d R c )
3520, 23, 19, 33, 34syl22anc 1219 . . 3  |-  ( ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  /\  ( ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A ) )  /\  a  =/=  b ) )  ->  E. c  e.  dom  ( a  \  b
) A. d  e. 
dom  ( a  \ 
b )  -.  d R c )
362, 3, 4, 6, 7, 35wemapsolem 7764 . 2  |-  ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  ->  T  Or  ( B  ^m  A ) )
371, 36syl3an1 1251 1  |-  ( ( A  e.  V  /\  R  We  A  /\  S  Or  B )  ->  T  Or  ( B  ^m  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   E.wrex 2716   _Vcvv 2972    \ cdif 3325    C_ wss 3328   (/)c0 3637   class class class wbr 4292   {copab 4349    Or wor 4640    Fr wfr 4676    We wwe 4678   dom cdm 4840    Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091    ^m cmap 7214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-map 7216
This theorem is referenced by:  opsrtoslem2  17566  wepwso  29395
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