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Theorem wemapso 7512
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 2956 . 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 3359 . . 3  |-  ( B  ^m  A )  C_  ( B  ^m  A )
4 simp1 957 . . 3  |-  ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  ->  A  e.  _V )
5 weso 4565 . . . 4  |-  ( R  We  A  ->  R  Or  A )
653ad2ant2 979 . . 3  |-  ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  ->  R  Or  A )
7 simp3 959 . . 3  |-  ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  ->  S  Or  B )
8 simpl1 960 . . . . 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 3466 . . . . . . 7  |-  ( a 
\  b )  C_  a
10 dmss 5061 . . . . . . 7  |-  ( ( a  \  b ) 
C_  a  ->  dom  ( a  \  b
)  C_  dom  a )
119, 10ax-mp 8 . . . . . 6  |-  dom  (
a  \  b )  C_ 
dom  a
12 simprll 739 . . . . . . . . 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 7030 . . . . . . . . 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 5583 . . . . . . . 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 5536 . . . . . . 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 3388 . . . . 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 4342 . . . 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 961 . . . . 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 4564 . . . . 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 734 . . . . 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 740 . . . . . . . . 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 7030 . . . . . . . . 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 5583 . . . . . . . 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 5828 . . . . . . 7  |-  ( ( a  Fn  A  /\  b  Fn  A )  ->  ( dom  ( a 
\  b )  =  (/) 
<->  a  =  b ) )
3116, 29, 30syl2anc 643 . . . . . 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 2633 . . . . 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 224 . . . 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 4536 . . . 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 1185 . . 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, 35wemapso2lem 7511 . 2  |-  ( ( A  e.  _V  /\  R  We  A  /\  S  Or  B )  ->  T  Or  ( B  ^m  A ) )
371, 36syl3an1 1217 1  |-  ( ( A  e.  V  /\  R  We  A  /\  S  Or  B )  ->  T  Or  ( B  ^m  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948    \ cdif 3309    C_ wss 3312   (/)c0 3620   class class class wbr 4204   {copab 4257    Or wor 4494    Fr wfr 4530    We wwe 4532   dom cdm 4870    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    ^m cmap 7010
This theorem is referenced by:  opsrtoslem2  16537  wepwso  27108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-map 7012
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