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Mirrors  >  Home  >  MPE Home  >  Th. List  >  r0weon Structured version   Unicode version
Theorem r0weon 8182
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 5694 . . . . . . . . . . . 12 |- ( x = z -> ( 1st ` x ) = ( 1st ` z ) )
3 fveq2 5694 . . . . . . . . . . . 12 |- ( x = z -> ( 2nd ` x ) = ( 2nd ` z ) )
42, 3uneq12d 3514 . . . . . . . . . . 11 |- ( x = z -> ( ( 1st ` x ) u. ( 2nd ` x ) ) = ( ( 1st ` z ) u. ( 2nd ` z ) ) )
5 eqid 2443 . . . . . . . . . . 11 |- ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) = ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) )
6 fvex 5704 . . . . . . . . . . . 12 |- ( 1st ` z ) e. _V
7 fvex 5704 . . . . . . . . . . . 12 |- ( 2nd ` z ) e. _V
86, 7unex 6381 . . . . . . . . . . 11 |- ( ( 1st ` z ) u. ( 2nd ` z ) ) e. _V
94, 5, 8fvmpt 5777 . . . . . . . . . 10 |- ( z e. ( On X. On ) -> ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` z ) = ( ( 1st ` z ) u. ( 2nd ` z ) ) )
10 fveq2 5694 . . . . . . . . . . . 12 |- ( x = w -> ( 1st ` x ) = ( 1st ` w ) )
11 fveq2 5694 . . . . . . . . . . . 12 |- ( x = w -> ( 2nd ` x ) = ( 2nd ` w ) )
1210, 11uneq12d 3514 . . . . . . . . . . 11 |- ( x = w -> ( ( 1st ` x ) u. ( 2nd ` x ) ) = ( ( 1st ` w ) u. ( 2nd ` w ) ) )
13 fvex 5704 . . . . . . . . . . . 12 |- ( 1st ` w ) e. _V
14 fvex 5704 . . . . . . . . . . . 12 |- ( 2nd ` w ) e. _V
1513, 14unex 6381 . . . . . . . . . . 11 |- ( ( 1st ` w ) u. ( 2nd ` w ) ) e. _V
1612, 5, 15fvmpt 5777 . . . . . . . . . 10 |- ( w e. ( On X. On ) -> ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` w ) = ( ( 1st ` w ) u. ( 2nd ` w ) ) )
179, 16breqan12d 4310 . . . . . . . . 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 4637 . . . . . . . . 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 2458 . . . . . . . . 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 704 . . . . . . . 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 709 . . . . . . 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 637 . . . . . 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 4359 . . . . 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 2466 . . . 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 6609 . . . . . . . 8 |- ( x e. ( On X. On ) -> ( 1st ` x ) e. On )
27 xp2nd 6610 . . . . . . . 8 |- ( x e. ( On X. On ) -> ( 2nd ` x ) e. On )
28 fvex 5704 . . . . . . . . . 10 |- ( 1st ` x ) e. _V
2928elon 4731 . . . . . . . . 9 |- ( ( 1st ` x ) e. On <-> Ord ( 1st ` x ) )
30 fvex 5704 . . . . . . . . . 10 |- ( 2nd ` x ) e. _V
3130elon 4731 . . . . . . . . 9 |- ( ( 2nd ` x ) e. On <-> Ord ( 2nd ` x ) )
32 ordun 4823 . . . . . . . . 9 |- ( ( Ord ( 1st ` x ) /\ Ord ( 2nd ` x ) ) -> Ord ( ( 1st ` x ) u. ( 2nd ` x ) ) )
3329, 31, 32syl2anb 479 . . . . . . . 8 |- ( ( ( 1st ` x ) e. On /\ ( 2nd ` x ) e. On ) -> Ord ( ( 1st ` x ) u. ( 2nd ` x ) ) )
3426, 27, 33syl2anc 661 . . . . . . 7 |- ( x e. ( On X. On ) -> Ord ( ( 1st ` x ) u. ( 2nd ` x ) ) )
3528, 30unex 6381 . . . . . . . 8 |- ( ( 1st ` x ) u. ( 2nd ` x ) ) e. _V
3635elon 4731 . . . . . . 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 5869 . . . . 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 6398 . . . . 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 8181 . . . . 5 |- L We ( On X. On )
4443a1i 11 . . . 4 |- ( T. -> L We ( On X. On ) )
45 vex 2978 . . . . . . . 8 |- u e. _V
4645dmex 6514 . . . . . . 7 |- dom u e. _V
4745rnex 6515 . . . . . . 7 |- ran u e. _V
4846, 47unex 6381 . . . . . 6 |- ( dom u u. ran u ) e. _V
49 imadmres 5333 . . . . . . 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 3574 . . . . . . . . . 10 |- ( u i^i ( On X. On ) ) C_ ( On X. On )
51 ssun1 3522 . . . . . . . . . . . . . 14 |- dom u C_ ( dom u u. ran u )
5250sseli 3355 . . . . . . . . . . . . . . . . 17 |- ( x e. ( u i^i ( On X. On ) ) -> x e. ( On X. On ) )
53 1st2nd2 6616 . . . . . . . . . . . . . . . . 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 3573 . . . . . . . . . . . . . . . . 17 |- ( u i^i ( On X. On ) ) C_ u
5655sseli 3355 . . . . . . . . . . . . . . . 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 5046 . . . . . . . . . . . . . . 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 3357 . . . . . . . . . . . . 13 |- ( x e. ( u i^i ( On X. On ) ) -> ( 1st ` x ) e. ( dom u u. ran u ) )
61 ssun2 3523 . . . . . . . . . . . . . 14 |- ran u C_ ( dom u u. ran u )
6228, 30opelrn 5074 . . . . . . . . . . . . . . 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 3357 . . . . . . . . . . . . 13 |- ( x e. ( u i^i ( On X. On ) ) -> ( 2nd ` x ) e. ( dom u u. ran u ) )
65 prssi 4032 . . . . . . . . . . . . 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 661 . . . . . . . . . . . 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 6440 . . . . . . . . . . . . 13 |- ( ( ( 1st ` x ) e. On /\ ( 2nd ` x ) e. On ) -> ( ( 1st ` x ) u. ( 2nd ` x ) ) e. { ( 1st ` x ) , ( 2nd ` x ) } )
7067, 68, 69syl2anc 661 . . . . . . . . . . . 12 |- ( x e. ( u i^i ( On X. On ) ) -> ( ( 1st ` x ) u. ( 2nd ` x ) ) e. { ( 1st ` x ) , ( 2nd ` x ) } )
7166, 70sseldd 3360 . . . . . . . . . . 11 |- ( x e. ( u i^i ( On X. On ) ) -> ( ( 1st ` x ) u. ( 2nd ` x ) ) e. ( dom u u. ran u ) )
7271rgen 2784 . . . . . . . . . 10 |- A. x e. ( u i^i ( On X. On ) ) ( ( 1st ` x ) u. ( 2nd ` x ) ) e. ( dom u u. ran u )
73 ssrab 3433 . . . . . . . . . 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 911 . . . . . . . . 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 5134 . . . . . . . . . 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 5567 . . . . . . . . . . 11 |- dom ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) = ( On X. On )
7776ineq2i 3552 . . . . . . . . . 10 |- ( u i^i dom ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ) = ( u i^i ( On X. On ) )
7875, 77eqtri 2463 . . . . . . . . 9 |- dom ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |` u ) = ( u i^i ( On X. On ) )
795mptpreima 5334 . . . . . . . . 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 3398 . . . . . . . 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 5457 . . . . . . . . 9 |- Fun ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) )
82 resss 5137 . . . . . . . . . 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 5042 . . . . . . . . . 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 5822 . . . . . . . . 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 672 . . . . . . . 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 3390 . . . . . 6 |- ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " u ) C_ ( dom u u. ran u )
8948, 88ssexi 4440 . . . . 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 6691 . . 3 |- ( T. -> R We ( On X. On ) )
92 epse 4706 . . . . 5 |- _E Se On
9392a1i 11 . . . 4 |- ( T. -> _E Se On )
9445uniex 6379 . . . . . . . 8 |- U. u e. _V
9594pwex 4478 . . . . . . 7 |- ~P U. u e. _V
9695, 95xpex 6511 . . . . . 6 |- ( ~P U. u X. ~P U. u ) e. _V
975mptpreima 5334 . . . . . . . 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 2727 . . . . . . . 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 2463 . . . . . . 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 465 . . . . . . . . 9 |- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
101 elssuni 4124 . . . . . . . . . . . . 13 |- ( ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u -> ( ( 1st ` x ) u. ( 2nd ` x ) ) C_ U. u )
102101adantl 466 . . . . . . . . . . . 12 |- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> ( ( 1st ` x ) u. ( 2nd ` x ) ) C_ U. u )
103102unssad 3536 . . . . . . . . . . 11 |- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> ( 1st ` x ) C_ U. u )
10428elpw 3869 . . . . . . . . . . 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 3537 . . . . . . . . . . 11 |- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> ( 2nd ` x ) C_ U. u )
10730elpw 3869 . . . . . . . . . . 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 532 . . . . . . . . 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 6611 . . . . . . . . 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 664 . . . . . . . 8 |- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> x e. ( ~P U. u X. ~P U. u ) )
112111abssi 3430 . . . . . . 7 |- { x | ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) } C_ ( ~P U. u X. ~P U. u )
11399, 112eqsstri 3389 . . . . . 6 |- ( `' ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " u ) C_ ( ~P U. u X. ~P U. u )
11496, 113ssexi 4440 . . . . 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 6692 . . 3 |- ( T. -> R Se ( On X. On ) )
11791, 116jca 532 . 2 |- ( T. -> ( R We ( On X. On ) /\ R Se ( On X. On ) ) )
118117trud 1378 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 1369   T. wtru 1370   e. wcel 1756   {cab 2429   A.wral 2718   {crab 2722   _Vcvv 2975   u. cun 3329   i^i cin 3330   C_ wss 3331   ~Pcpw 3863   {cpr 3882   <.cop 3886   U.cuni 4094   class class class wbr 4295   {copab 4352   e. cmpt 4353   _E cep 4633   Se wse 4680   We wwe 4681   Ord word 4721   Oncon0 4722   X. cxp 4841   `'ccnv 4842   dom cdm 4843   ran crn 4844   |` cres 4845   "cima 4846   Fun wfun 5415   -->wf 5417   ` cfv 5421   1stc1st 6578   2ndc2nd 6579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-rab 2727  df-v 2977  df-sbc 3190  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-1st 6580  df-2nd 6581
This theorem is referenced by:  infxpenlem  8183
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