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Theorem r0weon 8175
Description: A set-like well-ordering of the class of ordinal pairs. Proposition 7.58(1) of [TakeutiZaring] p. 54. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
leweon.1  |-  L  =  { <. x ,  y
>.  |  ( (
x  e.  ( On 
X.  On )  /\  y  e.  ( On  X.  On ) )  /\  ( ( 1st `  x
)  e.  ( 1st `  y )  \/  (
( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x )  e.  ( 2nd `  y
) ) ) ) }
r0weon.1  |-  R  =  { <. z ,  w >.  |  ( ( z  e.  ( On  X.  On )  /\  w  e.  ( On  X.  On ) )  /\  (
( ( 1st `  z
)  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  \/  ( ( ( 1st `  z )  u.  ( 2nd `  z ) )  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )  /\  z L w ) ) ) }
Assertion
Ref Expression
r0weon  |-  ( R  We  ( On  X.  On )  /\  R Se  ( On  X.  On ) )
Distinct variable groups:    z, w, L    x, w, y, z
Allowed substitution hints:    R( x, y, z, w)    L( x, y)

Proof of Theorem r0weon
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 r0weon.1 . . . . 5  |-  R  =  { <. z ,  w >.  |  ( ( z  e.  ( On  X.  On )  /\  w  e.  ( On  X.  On ) )  /\  (
( ( 1st `  z
)  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  \/  ( ( ( 1st `  z )  u.  ( 2nd `  z ) )  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )  /\  z L w ) ) ) }
2 fveq2 5688 . . . . . . . . . . . 12  |-  ( x  =  z  ->  ( 1st `  x )  =  ( 1st `  z
) )
3 fveq2 5688 . . . . . . . . . . . 12  |-  ( x  =  z  ->  ( 2nd `  x )  =  ( 2nd `  z
) )
42, 3uneq12d 3508 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
( 1st `  x
)  u.  ( 2nd `  x ) )  =  ( ( 1st `  z
)  u.  ( 2nd `  z ) ) )
5 eqid 2441 . . . . . . . . . . 11  |-  ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x
) ) )  =  ( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
6 fvex 5698 . . . . . . . . . . . 12  |-  ( 1st `  z )  e.  _V
7 fvex 5698 . . . . . . . . . . . 12  |-  ( 2nd `  z )  e.  _V
86, 7unex 6377 . . . . . . . . . . 11  |-  ( ( 1st `  z )  u.  ( 2nd `  z
) )  e.  _V
94, 5, 8fvmpt 5771 . . . . . . . . . 10  |-  ( z  e.  ( On  X.  On )  ->  ( ( x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) ) `
 z )  =  ( ( 1st `  z
)  u.  ( 2nd `  z ) ) )
10 fveq2 5688 . . . . . . . . . . . 12  |-  ( x  =  w  ->  ( 1st `  x )  =  ( 1st `  w
) )
11 fveq2 5688 . . . . . . . . . . . 12  |-  ( x  =  w  ->  ( 2nd `  x )  =  ( 2nd `  w
) )
1210, 11uneq12d 3508 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
( 1st `  x
)  u.  ( 2nd `  x ) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) )
13 fvex 5698 . . . . . . . . . . . 12  |-  ( 1st `  w )  e.  _V
14 fvex 5698 . . . . . . . . . . . 12  |-  ( 2nd `  w )  e.  _V
1513, 14unex 6377 . . . . . . . . . . 11  |-  ( ( 1st `  w )  u.  ( 2nd `  w
) )  e.  _V
1612, 5, 15fvmpt 5771 . . . . . . . . . 10  |-  ( w  e.  ( On  X.  On )  ->  ( ( x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) ) `
 w )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) )
179, 16breqan12d 4304 . . . . . . . . 9  |-  ( ( z  e.  ( On 
X.  On )  /\  w  e.  ( On  X.  On ) )  -> 
( ( ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x
) ) ) `  z )  _E  (
( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) ) `
 w )  <->  ( ( 1st `  z )  u.  ( 2nd `  z
) )  _E  (
( 1st `  w
)  u.  ( 2nd `  w ) ) ) )
1815epelc 4630 . . . . . . . . 9  |-  ( ( ( 1st `  z
)  u.  ( 2nd `  z ) )  _E  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  <->  ( ( 1st `  z )  u.  ( 2nd `  z
) )  e.  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) )
1917, 18syl6bb 261 . . . . . . . 8  |-  ( ( z  e.  ( On 
X.  On )  /\  w  e.  ( On  X.  On ) )  -> 
( ( ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x
) ) ) `  z )  _E  (
( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) ) `
 w )  <->  ( ( 1st `  z )  u.  ( 2nd `  z
) )  e.  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) ) )
209, 16eqeqan12d 2456 . . . . . . . . 9  |-  ( ( z  e.  ( On 
X.  On )  /\  w  e.  ( On  X.  On ) )  -> 
( ( ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x
) ) ) `  z )  =  ( ( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) ) `
 w )  <->  ( ( 1st `  z )  u.  ( 2nd `  z
) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) ) )
2120anbi1d 699 . . . . . . . 8  |-  ( ( z  e.  ( On 
X.  On )  /\  w  e.  ( On  X.  On ) )  -> 
( ( ( ( x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) ) `
 z )  =  ( ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x ) ) ) `  w )  /\  z L w )  <->  ( ( ( 1st `  z )  u.  ( 2nd `  z
) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  /\  z L w ) ) )
2219, 21orbi12d 704 . . . . . . 7  |-  ( ( z  e.  ( On 
X.  On )  /\  w  e.  ( On  X.  On ) )  -> 
( ( ( ( x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) ) `
 z )  _E  ( ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x ) ) ) `  w )  \/  ( ( ( x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) ) `
 z )  =  ( ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x ) ) ) `  w )  /\  z L w ) )  <->  ( (
( 1st `  z
)  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  \/  ( ( ( 1st `  z )  u.  ( 2nd `  z ) )  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )  /\  z L w ) ) ) )
2322pm5.32i 632 . . . . . 6  |-  ( ( ( z  e.  ( On  X.  On )  /\  w  e.  ( On  X.  On ) )  /\  ( ( ( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) ) `
 z )  _E  ( ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x ) ) ) `  w )  \/  ( ( ( x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) ) `
 z )  =  ( ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x ) ) ) `  w )  /\  z L w ) ) )  <->  ( (
z  e.  ( On 
X.  On )  /\  w  e.  ( On  X.  On ) )  /\  ( ( ( 1st `  z )  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w )  u.  ( 2nd `  w ) )  \/  ( ( ( 1st `  z )  u.  ( 2nd `  z
) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  /\  z L w ) ) ) )
2423opabbii 4353 . . . . 5  |-  { <. z ,  w >.  |  ( ( z  e.  ( On  X.  On )  /\  w  e.  ( On  X.  On ) )  /\  ( ( ( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) ) `
 z )  _E  ( ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x ) ) ) `  w )  \/  ( ( ( x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) ) `
 z )  =  ( ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x ) ) ) `  w )  /\  z L w ) ) ) }  =  { <. z ,  w >.  |  (
( z  e.  ( On  X.  On )  /\  w  e.  ( On  X.  On ) )  /\  ( ( ( 1st `  z
)  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  \/  ( ( ( 1st `  z )  u.  ( 2nd `  z ) )  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )  /\  z L w ) ) ) }
251, 24eqtr4i 2464 . . . 4  |-  R  =  { <. z ,  w >.  |  ( ( z  e.  ( On  X.  On )  /\  w  e.  ( On  X.  On ) )  /\  (
( ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x ) ) ) `  z )  _E  ( ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x
) ) ) `  w )  \/  (
( ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x ) ) ) `  z )  =  ( ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x
) ) ) `  w )  /\  z L w ) ) ) }
26 xp1st 6605 . . . . . . . 8  |-  ( x  e.  ( On  X.  On )  ->  ( 1st `  x )  e.  On )
27 xp2nd 6606 . . . . . . . 8  |-  ( x  e.  ( On  X.  On )  ->  ( 2nd `  x )  e.  On )
28 fvex 5698 . . . . . . . . . 10  |-  ( 1st `  x )  e.  _V
2928elon 4724 . . . . . . . . 9  |-  ( ( 1st `  x )  e.  On  <->  Ord  ( 1st `  x ) )
30 fvex 5698 . . . . . . . . . 10  |-  ( 2nd `  x )  e.  _V
3130elon 4724 . . . . . . . . 9  |-  ( ( 2nd `  x )  e.  On  <->  Ord  ( 2nd `  x ) )
32 ordun 4816 . . . . . . . . 9  |-  ( ( Ord  ( 1st `  x
)  /\  Ord  ( 2nd `  x ) )  ->  Ord  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
3329, 31, 32syl2anb 476 . . . . . . . 8  |-  ( ( ( 1st `  x
)  e.  On  /\  ( 2nd `  x )  e.  On )  ->  Ord  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
3426, 27, 33syl2anc 656 . . . . . . 7  |-  ( x  e.  ( On  X.  On )  ->  Ord  (
( 1st `  x
)  u.  ( 2nd `  x ) ) )
3528, 30unex 6377 . . . . . . . 8  |-  ( ( 1st `  x )  u.  ( 2nd `  x
) )  e.  _V
3635elon 4724 . . . . . . 7  |-  ( ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  On  <->  Ord  ( ( 1st `  x )  u.  ( 2nd `  x ) ) )
3734, 36sylibr 212 . . . . . 6  |-  ( x  e.  ( On  X.  On )  ->  ( ( 1st `  x )  u.  ( 2nd `  x
) )  e.  On )
385, 37fmpti 5863 . . . . 5  |-  ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x
) ) ) : ( On  X.  On )
--> On
3938a1i 11 . . . 4  |-  ( T. 
->  ( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) ) : ( On  X.  On ) --> On )
40 epweon 6394 . . . . 5  |-  _E  We  On
4140a1i 11 . . . 4  |-  ( T. 
->  _E  We  On )
42 leweon.1 . . . . . 6  |-  L  =  { <. x ,  y
>.  |  ( (
x  e.  ( On 
X.  On )  /\  y  e.  ( On  X.  On ) )  /\  ( ( 1st `  x
)  e.  ( 1st `  y )  \/  (
( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x )  e.  ( 2nd `  y
) ) ) ) }
4342leweon 8174 . . . . 5  |-  L  We  ( On  X.  On )
4443a1i 11 . . . 4  |-  ( T. 
->  L  We  ( On  X.  On ) )
45 vex 2973 . . . . . . . 8  |-  u  e. 
_V
4645dmex 6510 . . . . . . 7  |-  dom  u  e.  _V
4745rnex 6511 . . . . . . 7  |-  ran  u  e.  _V
4846, 47unex 6377 . . . . . 6  |-  ( dom  u  u.  ran  u
)  e.  _V
49 imadmres 5327 . . . . . . 7  |-  ( ( x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
" dom  ( (
x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )  |`  u ) )  =  ( ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x ) ) ) " u )
50 inss2 3568 . . . . . . . . . 10  |-  ( u  i^i  ( On  X.  On ) )  C_  ( On  X.  On )
51 ssun1 3516 . . . . . . . . . . . . . 14  |-  dom  u  C_  ( dom  u  u. 
ran  u )
5250sseli 3349 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  x  e.  ( On  X.  On ) )
53 1st2nd2 6612 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( On  X.  On )  ->  x  = 
<. ( 1st `  x
) ,  ( 2nd `  x ) >. )
5452, 53syl 16 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
55 inss1 3567 . . . . . . . . . . . . . . . . 17  |-  ( u  i^i  ( On  X.  On ) )  C_  u
5655sseli 3349 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  x  e.  u )
5754, 56eqeltrrd 2516 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  e.  u )
5828, 30opeldm 5039 . . . . . . . . . . . . . . 15  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  u  ->  ( 1st `  x
)  e.  dom  u
)
5957, 58syl 16 . . . . . . . . . . . . . 14  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  ( 1st `  x )  e. 
dom  u )
6051, 59sseldi 3351 . . . . . . . . . . . . 13  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  ( 1st `  x )  e.  ( dom  u  u. 
ran  u ) )
61 ssun2 3517 . . . . . . . . . . . . . 14  |-  ran  u  C_  ( dom  u  u. 
ran  u )
6228, 30opelrn 5067 . . . . . . . . . . . . . . 15  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  u  ->  ( 2nd `  x
)  e.  ran  u
)
6357, 62syl 16 . . . . . . . . . . . . . 14  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  ( 2nd `  x )  e. 
ran  u )
6461, 63sseldi 3351 . . . . . . . . . . . . 13  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  ( 2nd `  x )  e.  ( dom  u  u. 
ran  u ) )
65 prssi 4026 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  x
)  e.  ( dom  u  u.  ran  u
)  /\  ( 2nd `  x )  e.  ( dom  u  u.  ran  u ) )  ->  { ( 1st `  x
) ,  ( 2nd `  x ) }  C_  ( dom  u  u.  ran  u ) )
6660, 64, 65syl2anc 656 . . . . . . . . . . . 12  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  { ( 1st `  x ) ,  ( 2nd `  x
) }  C_  ( dom  u  u.  ran  u
) )
6752, 26syl 16 . . . . . . . . . . . . 13  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  ( 1st `  x )  e.  On )
6852, 27syl 16 . . . . . . . . . . . . 13  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  ( 2nd `  x )  e.  On )
69 ordunpr 6436 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  x
)  e.  On  /\  ( 2nd `  x )  e.  On )  -> 
( ( 1st `  x
)  u.  ( 2nd `  x ) )  e. 
{ ( 1st `  x
) ,  ( 2nd `  x ) } )
7067, 68, 69syl2anc 656 . . . . . . . . . . . 12  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  (
( 1st `  x
)  u.  ( 2nd `  x ) )  e. 
{ ( 1st `  x
) ,  ( 2nd `  x ) } )
7166, 70sseldd 3354 . . . . . . . . . . 11  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  (
( 1st `  x
)  u.  ( 2nd `  x ) )  e.  ( dom  u  u. 
ran  u ) )
7271rgen 2779 . . . . . . . . . 10  |-  A. x  e.  ( u  i^i  ( On  X.  On ) ) ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  ( dom  u  u. 
ran  u )
73 ssrab 3427 . . . . . . . . . 10  |-  ( ( u  i^i  ( On 
X.  On ) ) 
C_  { x  e.  ( On  X.  On )  |  ( ( 1st `  x )  u.  ( 2nd `  x
) )  e.  ( dom  u  u.  ran  u ) }  <->  ( (
u  i^i  ( On  X.  On ) )  C_  ( On  X.  On )  /\  A. x  e.  ( u  i^i  ( On  X.  On ) ) ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  ( dom  u  u. 
ran  u ) ) )
7450, 72, 73mpbir2an 906 . . . . . . . . 9  |-  ( u  i^i  ( On  X.  On ) )  C_  { x  e.  ( On  X.  On )  |  ( ( 1st `  x )  u.  ( 2nd `  x
) )  e.  ( dom  u  u.  ran  u ) }
75 dmres 5128 . . . . . . . . . 10  |-  dom  (
( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )  |`  u )  =  ( u  i^i  dom  (
x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) ) )
7638fdmi 5561 . . . . . . . . . . 11  |-  dom  (
x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )  =  ( On  X.  On )
7776ineq2i 3546 . . . . . . . . . 10  |-  ( u  i^i  dom  ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x ) ) ) )  =  ( u  i^i  ( On 
X.  On ) )
7875, 77eqtri 2461 . . . . . . . . 9  |-  dom  (
( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )  |`  u )  =  ( u  i^i  ( On 
X.  On ) )
795mptpreima 5328 . . . . . . . . 9  |-  ( `' ( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
" ( dom  u  u.  ran  u ) )  =  { x  e.  ( On  X.  On )  |  ( ( 1st `  x )  u.  ( 2nd `  x
) )  e.  ( dom  u  u.  ran  u ) }
8074, 78, 793sstr4i 3392 . . . . . . . 8  |-  dom  (
( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )  |`  u )  C_  ( `' ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x ) ) ) " ( dom  u  u.  ran  u
) )
81 funmpt 5451 . . . . . . . . 9  |-  Fun  (
x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
82 resss 5131 . . . . . . . . . 10  |-  ( ( x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )  |`  u )  C_  (
x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
83 dmss 5035 . . . . . . . . . 10  |-  ( ( ( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )  |`  u )  C_  (
x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )  ->  dom  ( (
x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )  |`  u )  C_  dom  ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) ) )
8482, 83ax-mp 5 . . . . . . . . 9  |-  dom  (
( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )  |`  u )  C_  dom  ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
85 funimass3 5816 . . . . . . . . 9  |-  ( ( Fun  ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x ) ) )  /\  dom  (
( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )  |`  u )  C_  dom  ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) ) )  ->  ( (
( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
" dom  ( (
x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )  |`  u ) )  C_  ( dom  u  u.  ran  u )  <->  dom  ( ( x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )  |`  u )  C_  ( `' ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x ) ) ) " ( dom  u  u.  ran  u
) ) ) )
8681, 84, 85mp2an 667 . . . . . . . 8  |-  ( ( ( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
" dom  ( (
x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )  |`  u ) )  C_  ( dom  u  u.  ran  u )  <->  dom  ( ( x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )  |`  u )  C_  ( `' ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x ) ) ) " ( dom  u  u.  ran  u
) ) )
8780, 86mpbir 209 . . . . . . 7  |-  ( ( x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
" dom  ( (
x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )  |`  u ) )  C_  ( dom  u  u.  ran  u )
8849, 87eqsstr3i 3384 . . . . . 6  |-  ( ( x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
" u )  C_  ( dom  u  u.  ran  u )
8948, 88ssexi 4434 . . . . 5  |-  ( ( x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
" u )  e. 
_V
9089a1i 11 . . . 4  |-  ( T. 
->  ( ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x ) ) ) " u )  e.  _V )
9125, 39, 41, 44, 90fnwe 6687 . . 3  |-  ( T. 
->  R  We  ( On  X.  On ) )
92 epse 4699 . . . . 5  |-  _E Se  On
9392a1i 11 . . . 4  |-  ( T. 
->  _E Se  On )
9445uniex 6375 . . . . . . . 8  |-  U. u  e.  _V
9594pwex 4472 . . . . . . 7  |-  ~P U. u  e.  _V
9695, 95xpex 6507 . . . . . 6  |-  ( ~P
U. u  X.  ~P U. u )  e.  _V
975mptpreima 5328 . . . . . . . 8  |-  ( `' ( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
" u )  =  { x  e.  ( On  X.  On )  |  ( ( 1st `  x )  u.  ( 2nd `  x ) )  e.  u }
98 df-rab 2722 . . . . . . . 8  |-  { x  e.  ( On  X.  On )  |  ( ( 1st `  x )  u.  ( 2nd `  x
) )  e.  u }  =  { x  |  ( x  e.  ( On  X.  On )  /\  ( ( 1st `  x )  u.  ( 2nd `  x ) )  e.  u ) }
9997, 98eqtri 2461 . . . . . . 7  |-  ( `' ( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
" u )  =  { x  |  ( x  e.  ( On 
X.  On )  /\  ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  u ) }
10053adantr 462 . . . . . . . . 9  |-  ( ( x  e.  ( On 
X.  On )  /\  ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  u )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
101 elssuni 4118 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  u  ->  ( ( 1st `  x )  u.  ( 2nd `  x
) )  C_  U. u
)
102101adantl 463 . . . . . . . . . . . 12  |-  ( ( x  e.  ( On 
X.  On )  /\  ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  u )  ->  (
( 1st `  x
)  u.  ( 2nd `  x ) )  C_  U. u )
103102unssad 3530 . . . . . . . . . . 11  |-  ( ( x  e.  ( On 
X.  On )  /\  ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  u )  ->  ( 1st `  x )  C_  U. u )
10428elpw 3863 . . . . . . . . . . 11  |-  ( ( 1st `  x )  e.  ~P U. u  <->  ( 1st `  x ) 
C_  U. u )
105103, 104sylibr 212 . . . . . . . . . 10  |-  ( ( x  e.  ( On 
X.  On )  /\  ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  u )  ->  ( 1st `  x )  e. 
~P U. u )
106102unssbd 3531 . . . . . . . . . . 11  |-  ( ( x  e.  ( On 
X.  On )  /\  ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  u )  ->  ( 2nd `  x )  C_  U. u )
10730elpw 3863 . . . . . . . . . . 11  |-  ( ( 2nd `  x )  e.  ~P U. u  <->  ( 2nd `  x ) 
C_  U. u )
108106, 107sylibr 212 . . . . . . . . . 10  |-  ( ( x  e.  ( On 
X.  On )  /\  ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  u )  ->  ( 2nd `  x )  e. 
~P U. u )
109105, 108jca 529 . . . . . . . . 9  |-  ( ( x  e.  ( On 
X.  On )  /\  ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  u )  ->  (
( 1st `  x
)  e.  ~P U. u  /\  ( 2nd `  x
)  e.  ~P U. u ) )
110 elxp6 6607 . . . . . . . . 9  |-  ( x  e.  ( ~P U. u  X.  ~P U. u
)  <->  ( x  = 
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  /\  (
( 1st `  x
)  e.  ~P U. u  /\  ( 2nd `  x
)  e.  ~P U. u ) ) )
111100, 109, 110sylanbrc 659 . . . . . . . 8  |-  ( ( x  e.  ( On 
X.  On )  /\  ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  u )  ->  x  e.  ( ~P U. u  X.  ~P U. u ) )
112111abssi 3424 . . . . . . 7  |-  { x  |  ( x  e.  ( On  X.  On )  /\  ( ( 1st `  x )  u.  ( 2nd `  x ) )  e.  u ) } 
C_  ( ~P U. u  X.  ~P U. u
)
11399, 112eqsstri 3383 . . . . . 6  |-  ( `' ( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
" u )  C_  ( ~P U. u  X.  ~P U. u )
11496, 113ssexi 4434 . . . . 5  |-  ( `' ( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
" u )  e. 
_V
115114a1i 11 . . . 4  |-  ( T. 
->  ( `' ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x
) ) ) "
u )  e.  _V )
11625, 39, 93, 115fnse 6688 . . 3  |-  ( T. 
->  R Se  ( On  X.  On ) )
11791, 116jca 529 . 2  |-  ( T. 
->  ( R  We  ( On  X.  On )  /\  R Se  ( On  X.  On ) ) )
118117trud 1373 1  |-  ( R  We  ( On  X.  On )  /\  R Se  ( On  X.  On ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1364   T. wtru 1365    e. wcel 1761   {cab 2427   A.wral 2713   {crab 2717   _Vcvv 2970    u. cun 3323    i^i cin 3324    C_ wss 3325   ~Pcpw 3857   {cpr 3876   <.cop 3880   U.cuni 4088   class class class wbr 4289   {copab 4346    e. cmpt 4347    _E cep 4626   Se wse 4673    We wwe 4674   Ord word 4714   Oncon0 4715    X. cxp 4834   `'ccnv 4835   dom cdm 4836   ran crn 4837    |` cres 4838   "cima 4839   Fun wfun 5409   -->wf 5411   ` cfv 5415   1stc1st 6574   2ndc2nd 6575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-1st 6576  df-2nd 6577
This theorem is referenced by:  infxpenlem  8176
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