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Theorem wemappo 7877
Description: Construct lexicographic order on a function space based on a well-ordering of the indexes and a total ordering of the values.

Without totality on the values or least differing indexes, the best we can prove here is a partial order. (Contributed by Stefan O'Rear, 18-Jan-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
wemappo  |-  ( ( A  e.  V  /\  R  Or  A  /\  S  Po  B )  ->  T  Po  ( 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 wemappo
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3087 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 simpll3 1029 . . . . . . 7  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  a  e.  ( B  ^m  A ) )  /\  b  e.  A )  ->  S  Po  B )
3 elmapi 7347 . . . . . . . . 9  |-  ( a  e.  ( B  ^m  A )  ->  a : A --> B )
43adantl 466 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Po  B )  /\  a  e.  ( B  ^m  A ) )  ->  a : A --> B )
54ffvelrnda 5955 . . . . . . 7  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  a  e.  ( B  ^m  A ) )  /\  b  e.  A )  ->  (
a `  b )  e.  B )
6 poirr 4763 . . . . . . 7  |-  ( ( S  Po  B  /\  ( a `  b
)  e.  B )  ->  -.  ( a `  b ) S ( a `  b ) )
72, 5, 6syl2anc 661 . . . . . 6  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  a  e.  ( B  ^m  A ) )  /\  b  e.  A )  ->  -.  ( a `  b
) S ( a `
 b ) )
87intnanrd 908 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  a  e.  ( B  ^m  A ) )  /\  b  e.  A )  ->  -.  ( ( a `  b ) S ( a `  b )  /\  A. c  e.  A  ( c R b  ->  ( a `  c )  =  ( a `  c ) ) ) )
98nrexdv 2925 . . . 4  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Po  B )  /\  a  e.  ( B  ^m  A ) )  ->  -.  E. b  e.  A  ( (
a `  b ) S ( a `  b )  /\  A. c  e.  A  (
c R b  -> 
( a `  c
)  =  ( a `
 c ) ) ) )
10 vex 3081 . . . . 5  |-  a  e. 
_V
11 wemapso.t . . . . . 6  |-  T  =  { <. x ,  y
>.  |  E. z  e.  A  ( (
x `  z ) S ( y `  z )  /\  A. w  e.  A  (
w R z  -> 
( x `  w
)  =  ( y `
 w ) ) ) }
1211wemaplem1 7874 . . . . 5  |-  ( ( a  e.  _V  /\  a  e.  _V )  ->  ( a T a  <->  E. b  e.  A  ( ( a `  b ) S ( a `  b )  /\  A. c  e.  A  ( c R b  ->  ( a `  c )  =  ( a `  c ) ) ) ) )
1310, 10, 12mp2an 672 . . . 4  |-  ( a T a  <->  E. b  e.  A  ( (
a `  b ) S ( a `  b )  /\  A. c  e.  A  (
c R b  -> 
( a `  c
)  =  ( a `
 c ) ) ) )
149, 13sylnibr 305 . . 3  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Po  B )  /\  a  e.  ( B  ^m  A ) )  ->  -.  a T
a )
15 simpll1 1027 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  /\  (
a T b  /\  b T c ) )  ->  A  e.  _V )
16 simplr1 1030 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  /\  (
a T b  /\  b T c ) )  ->  a  e.  ( B  ^m  A ) )
17 simplr2 1031 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  /\  (
a T b  /\  b T c ) )  ->  b  e.  ( B  ^m  A ) )
18 simplr3 1032 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  /\  (
a T b  /\  b T c ) )  ->  c  e.  ( B  ^m  A ) )
19 simpll2 1028 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  /\  (
a T b  /\  b T c ) )  ->  R  Or  A
)
20 simpll3 1029 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  /\  (
a T b  /\  b T c ) )  ->  S  Po  B
)
21 simprl 755 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  /\  (
a T b  /\  b T c ) )  ->  a T b )
22 simprr 756 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  /\  (
a T b  /\  b T c ) )  ->  b T c )
2311, 15, 16, 17, 18, 19, 20, 21, 22wemaplem3 7876 . . . 4  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  /\  (
a T b  /\  b T c ) )  ->  a T c )
2423ex 434 . . 3  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Po  B )  /\  ( a  e.  ( B  ^m  A )  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  ->  (
( a T b  /\  b T c )  ->  a T
c ) )
2514, 24ispod 4760 . 2  |-  ( ( A  e.  _V  /\  R  Or  A  /\  S  Po  B )  ->  T  Po  ( B  ^m  A ) )
261, 25syl3an1 1252 1  |-  ( ( A  e.  V  /\  R  Or  A  /\  S  Po  B )  ->  T  Po  ( B  ^m  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   E.wrex 2800   _Vcvv 3078   class class class wbr 4403   {copab 4460    Po wpo 4750    Or wor 4751   -->wf 5525   ` cfv 5529  (class class class)co 6203    ^m cmap 7327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-map 7329
This theorem is referenced by:  wemapsolem  7878
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