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Theorem r0weon 8442
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 5881 . . . . . . . . . . . 12  |-  ( x  =  z  ->  ( 1st `  x )  =  ( 1st `  z
) )
3 fveq2 5881 . . . . . . . . . . . 12  |-  ( x  =  z  ->  ( 2nd `  x )  =  ( 2nd `  z
) )
42, 3uneq12d 3627 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
( 1st `  x
)  u.  ( 2nd `  x ) )  =  ( ( 1st `  z
)  u.  ( 2nd `  z ) ) )
5 eqid 2429 . . . . . . . . . . 11  |-  ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x
) ) )  =  ( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
6 fvex 5891 . . . . . . . . . . . 12  |-  ( 1st `  z )  e.  _V
7 fvex 5891 . . . . . . . . . . . 12  |-  ( 2nd `  z )  e.  _V
86, 7unex 6603 . . . . . . . . . . 11  |-  ( ( 1st `  z )  u.  ( 2nd `  z
) )  e.  _V
94, 5, 8fvmpt 5964 . . . . . . . . . 10  |-  ( z  e.  ( On  X.  On )  ->  ( ( x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) ) `
 z )  =  ( ( 1st `  z
)  u.  ( 2nd `  z ) ) )
10 fveq2 5881 . . . . . . . . . . . 12  |-  ( x  =  w  ->  ( 1st `  x )  =  ( 1st `  w
) )
11 fveq2 5881 . . . . . . . . . . . 12  |-  ( x  =  w  ->  ( 2nd `  x )  =  ( 2nd `  w
) )
1210, 11uneq12d 3627 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
( 1st `  x
)  u.  ( 2nd `  x ) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) )
13 fvex 5891 . . . . . . . . . . . 12  |-  ( 1st `  w )  e.  _V
14 fvex 5891 . . . . . . . . . . . 12  |-  ( 2nd `  w )  e.  _V
1513, 14unex 6603 . . . . . . . . . . 11  |-  ( ( 1st `  w )  u.  ( 2nd `  w
) )  e.  _V
1612, 5, 15fvmpt 5964 . . . . . . . . . 10  |-  ( w  e.  ( On  X.  On )  ->  ( ( x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) ) `
 w )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) )
179, 16breqan12d 4441 . . . . . . . . 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 4767 . . . . . . . . 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 264 . . . . . . . 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 2452 . . . . . . . . 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 709 . . . . . . . 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 714 . . . . . . 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 641 . . . . . 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 4490 . . . . 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 2461 . . . 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 6837 . . . . . . . 8  |-  ( x  e.  ( On  X.  On )  ->  ( 1st `  x )  e.  On )
27 xp2nd 6838 . . . . . . . 8  |-  ( x  e.  ( On  X.  On )  ->  ( 2nd `  x )  e.  On )
28 fvex 5891 . . . . . . . . . 10  |-  ( 1st `  x )  e.  _V
2928elon 5451 . . . . . . . . 9  |-  ( ( 1st `  x )  e.  On  <->  Ord  ( 1st `  x ) )
30 fvex 5891 . . . . . . . . . 10  |-  ( 2nd `  x )  e.  _V
3130elon 5451 . . . . . . . . 9  |-  ( ( 2nd `  x )  e.  On  <->  Ord  ( 2nd `  x ) )
32 ordun 5543 . . . . . . . . 9  |-  ( ( Ord  ( 1st `  x
)  /\  Ord  ( 2nd `  x ) )  ->  Ord  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
3329, 31, 32syl2anb 481 . . . . . . . 8  |-  ( ( ( 1st `  x
)  e.  On  /\  ( 2nd `  x )  e.  On )  ->  Ord  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
3426, 27, 33syl2anc 665 . . . . . . 7  |-  ( x  e.  ( On  X.  On )  ->  Ord  (
( 1st `  x
)  u.  ( 2nd `  x ) ) )
3528, 30unex 6603 . . . . . . . 8  |-  ( ( 1st `  x )  u.  ( 2nd `  x
) )  e.  _V
3635elon 5451 . . . . . . 7  |-  ( ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  On  <->  Ord  ( ( 1st `  x )  u.  ( 2nd `  x ) ) )
3734, 36sylibr 215 . . . . . 6  |-  ( x  e.  ( On  X.  On )  ->  ( ( 1st `  x )  u.  ( 2nd `  x
) )  e.  On )
385, 37fmpti 6060 . . . . 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 6624 . . . . 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 8441 . . . . 5  |-  L  We  ( On  X.  On )
4443a1i 11 . . . 4  |-  ( T. 
->  L  We  ( On  X.  On ) )
45 vex 3090 . . . . . . . 8  |-  u  e. 
_V
4645dmex 6740 . . . . . . 7  |-  dom  u  e.  _V
4745rnex 6741 . . . . . . 7  |-  ran  u  e.  _V
4846, 47unex 6603 . . . . . 6  |-  ( dom  u  u.  ran  u
)  e.  _V
49 imadmres 5347 . . . . . . 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 3689 . . . . . . . . . 10  |-  ( u  i^i  ( On  X.  On ) )  C_  ( On  X.  On )
51 ssun1 3635 . . . . . . . . . . . . . 14  |-  dom  u  C_  ( dom  u  u. 
ran  u )
5250sseli 3466 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  x  e.  ( On  X.  On ) )
53 1st2nd2 6844 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( On  X.  On )  ->  x  = 
<. ( 1st `  x
) ,  ( 2nd `  x ) >. )
5452, 53syl 17 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
55 inss1 3688 . . . . . . . . . . . . . . . . 17  |-  ( u  i^i  ( On  X.  On ) )  C_  u
5655sseli 3466 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  x  e.  u )
5754, 56eqeltrrd 2518 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  e.  u )
5828, 30opeldm 5058 . . . . . . . . . . . . . . 15  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  u  ->  ( 1st `  x
)  e.  dom  u
)
5957, 58syl 17 . . . . . . . . . . . . . 14  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  ( 1st `  x )  e. 
dom  u )
6051, 59sseldi 3468 . . . . . . . . . . . . 13  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  ( 1st `  x )  e.  ( dom  u  u. 
ran  u ) )
61 ssun2 3636 . . . . . . . . . . . . . 14  |-  ran  u  C_  ( dom  u  u. 
ran  u )
6228, 30opelrn 5086 . . . . . . . . . . . . . . 15  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  u  ->  ( 2nd `  x
)  e.  ran  u
)
6357, 62syl 17 . . . . . . . . . . . . . 14  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  ( 2nd `  x )  e. 
ran  u )
6461, 63sseldi 3468 . . . . . . . . . . . . 13  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  ( 2nd `  x )  e.  ( dom  u  u. 
ran  u ) )
65 prssi 4159 . . . . . . . . . . . . 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 665 . . . . . . . . . . . 12  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  { ( 1st `  x ) ,  ( 2nd `  x
) }  C_  ( dom  u  u.  ran  u
) )
6752, 26syl 17 . . . . . . . . . . . . 13  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  ( 1st `  x )  e.  On )
6852, 27syl 17 . . . . . . . . . . . . 13  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  ( 2nd `  x )  e.  On )
69 ordunpr 6667 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  x
)  e.  On  /\  ( 2nd `  x )  e.  On )  -> 
( ( 1st `  x
)  u.  ( 2nd `  x ) )  e. 
{ ( 1st `  x
) ,  ( 2nd `  x ) } )
7067, 68, 69syl2anc 665 . . . . . . . . . . . 12  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  (
( 1st `  x
)  u.  ( 2nd `  x ) )  e. 
{ ( 1st `  x
) ,  ( 2nd `  x ) } )
7166, 70sseldd 3471 . . . . . . . . . . 11  |-  ( x  e.  ( u  i^i  ( On  X.  On ) )  ->  (
( 1st `  x
)  u.  ( 2nd `  x ) )  e.  ( dom  u  u. 
ran  u ) )
7271rgen 2792 . . . . . . . . . 10  |-  A. x  e.  ( u  i^i  ( On  X.  On ) ) ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  ( dom  u  u. 
ran  u )
73 ssrab 3545 . . . . . . . . . 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 928 . . . . . . . . 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 5145 . . . . . . . . . 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 5751 . . . . . . . . . . 11  |-  dom  (
x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )  =  ( On  X.  On )
7776ineq2i 3667 . . . . . . . . . 10  |-  ( u  i^i  dom  ( x  e.  ( On  X.  On )  |->  ( ( 1st `  x )  u.  ( 2nd `  x ) ) ) )  =  ( u  i^i  ( On 
X.  On ) )
7875, 77eqtri 2458 . . . . . . . . 9  |-  dom  (
( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )  |`  u )  =  ( u  i^i  ( On 
X.  On ) )
795mptpreima 5348 . . . . . . . . 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 3509 . . . . . . . 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 5637 . . . . . . . . 9  |-  Fun  (
x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
82 resss 5148 . . . . . . . . . 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 5054 . . . . . . . . . 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 6013 . . . . . . . . 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 676 . . . . . . . 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 212 . . . . . . 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 3501 . . . . . 6  |-  ( ( x  e.  ( On 
X.  On )  |->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
" u )  C_  ( dom  u  u.  ran  u )
8948, 88ssexi 4570 . . . . 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 6923 . . 3  |-  ( T. 
->  R  We  ( On  X.  On ) )
92 epse 4837 . . . . 5  |-  _E Se  On
9392a1i 11 . . . 4  |-  ( T. 
->  _E Se  On )
9445uniex 6601 . . . . . . . 8  |-  U. u  e.  _V
9594pwex 4608 . . . . . . 7  |-  ~P U. u  e.  _V
9695, 95xpex 6609 . . . . . 6  |-  ( ~P
U. u  X.  ~P U. u )  e.  _V
975mptpreima 5348 . . . . . . . 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 2791 . . . . . . . 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 2458 . . . . . . 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 466 . . . . . . . . 9  |-  ( ( x  e.  ( On 
X.  On )  /\  ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  u )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
101 elssuni 4251 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  u  ->  ( ( 1st `  x )  u.  ( 2nd `  x
) )  C_  U. u
)
102101adantl 467 . . . . . . . . . . . 12  |-  ( ( x  e.  ( On 
X.  On )  /\  ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  u )  ->  (
( 1st `  x
)  u.  ( 2nd `  x ) )  C_  U. u )
103102unssad 3649 . . . . . . . . . . 11  |-  ( ( x  e.  ( On 
X.  On )  /\  ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  u )  ->  ( 1st `  x )  C_  U. u )
10428elpw 3991 . . . . . . . . . . 11  |-  ( ( 1st `  x )  e.  ~P U. u  <->  ( 1st `  x ) 
C_  U. u )
105103, 104sylibr 215 . . . . . . . . . 10  |-  ( ( x  e.  ( On 
X.  On )  /\  ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  u )  ->  ( 1st `  x )  e. 
~P U. u )
106102unssbd 3650 . . . . . . . . . . 11  |-  ( ( x  e.  ( On 
X.  On )  /\  ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  u )  ->  ( 2nd `  x )  C_  U. u )
10730elpw 3991 . . . . . . . . . . 11  |-  ( ( 2nd `  x )  e.  ~P U. u  <->  ( 2nd `  x ) 
C_  U. u )
108106, 107sylibr 215 . . . . . . . . . 10  |-  ( ( x  e.  ( On 
X.  On )  /\  ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  u )  ->  ( 2nd `  x )  e. 
~P U. u )
109105, 108jca 534 . . . . . . . . 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 6839 . . . . . . . . 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 668 . . . . . . . 8  |-  ( ( x  e.  ( On 
X.  On )  /\  ( ( 1st `  x
)  u.  ( 2nd `  x ) )  e.  u )  ->  x  e.  ( ~P U. u  X.  ~P U. u ) )
112111abssi 3542 . . . . . . 7  |-  { x  |  ( x  e.  ( On  X.  On )  /\  ( ( 1st `  x )  u.  ( 2nd `  x ) )  e.  u ) } 
C_  ( ~P U. u  X.  ~P U. u
)
11399, 112eqsstri 3500 . . . . . 6  |-  ( `' ( x  e.  ( On  X.  On ) 
|->  ( ( 1st `  x
)  u.  ( 2nd `  x ) ) )
" u )  C_  ( ~P U. u  X.  ~P U. u )
11496, 113ssexi 4570 . . . . 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 6924 . . 3  |-  ( T. 
->  R Se  ( On  X.  On ) )
11791, 116jca 534 . 2  |-  ( T. 
->  ( R  We  ( On  X.  On )  /\  R Se  ( On  X.  On ) ) )
118117trud 1446 1  |-  ( R  We  ( On  X.  On )  /\  R Se  ( On  X.  On ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437   T. wtru 1438    e. wcel 1870   {cab 2414   A.wral 2782   {crab 2786   _Vcvv 3087    u. cun 3440    i^i cin 3441    C_ wss 3442   ~Pcpw 3985   {cpr 4004   <.cop 4008   U.cuni 4222   class class class wbr 4426   {copab 4483    |-> cmpt 4484    _E cep 4763   Se wse 4811    We wwe 4812    X. cxp 4852   `'ccnv 4853   dom cdm 4854   ran crn 4855    |` cres 4856   "cima 4857   Ord word 5441   Oncon0 5442   Fun wfun 5595   -->wf 5597   ` cfv 5601   1stc1st 6805   2ndc2nd 6806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ord 5445  df-on 5446  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-1st 6807  df-2nd 6808
This theorem is referenced by:  infxpenlem  8443
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