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Theorem wemapso 7988
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 3127 . 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 3528 . . 3  |-  ( B  ^m  A )  C_  ( B  ^m  A )
4 simp1 996 . . 3  |-  ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  ->  A  e.  _V )
5 weso 4876 . . . 4  |-  ( R  We  A  ->  R  Or  A )
653ad2ant2 1018 . . 3  |-  ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  ->  R  Or  A )
7 simp3 998 . . 3  |-  ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  ->  S  Or  B )
8 simpl1 999 . . . . 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 3636 . . . . . . 7  |-  ( a 
\  b )  C_  a
10 dmss 5208 . . . . . . 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 7452 . . . . . . . . 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 5737 . . . . . . . 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 5686 . . . . . . 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 3557 . . . . 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 4600 . . . 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 1000 . . . . 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 4875 . . . . 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 7452 . . . . . . . . 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 5737 . . . . . . . 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 5994 . . . . . . 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 2725 . . . . 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 4847 . . . 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 1229 . . 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 7987 . 2  |-  ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  ->  T  Or  ( B  ^m  A ) )
371, 36syl3an1 1261 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818   _Vcvv 3118    \ cdif 3478    C_ wss 3481   (/)c0 3790   class class class wbr 4453   {copab 4510    Or wor 4805    Fr wfr 4841    We wwe 4843   dom cdm 5005    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^m cmap 7432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-map 7434
This theorem is referenced by:  opsrtoslem2  18019  wepwso  30916
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