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Theorem wemappo 7977
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 3104 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 simpll3 1038 . . . . . . 7  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  a  e.  ( B  ^m  A ) )  /\  b  e.  A )  ->  S  Po  B )
3 elmapi 7442 . . . . . . . . 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 6016 . . . . . . 7  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  a  e.  ( B  ^m  A ) )  /\  b  e.  A )  ->  (
a `  b )  e.  B )
6 poirr 4801 . . . . . . 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 917 . . . . 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 2899 . . . 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 3098 . . . . 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 7974 . . . . 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 1036 . . . . 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 1039 . . . . 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 1040 . . . . 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 1041 . . . . 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 1037 . . . . 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 1038 . . . . 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 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 ) )  ->  a T b )
22 simprr 757 . . . . 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 7976 . . . 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 4798 . 2  |-  ( ( A  e.  _V  /\  R  Or  A  /\  S  Po  B )  ->  T  Po  ( B  ^m  A ) )
261, 25syl3an1 1262 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 974    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794   _Vcvv 3095   class class class wbr 4437   {copab 4494    Po wpo 4788    Or wor 4789   -->wf 5574   ` cfv 5578  (class class class)co 6281    ^m cmap 7422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-map 7424
This theorem is referenced by:  wemapsolem  7978
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