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Theorem r0weon 8468
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 5887 . . . . . . . . . . . 12 |- ( x = z -> ( 1st ` x ) = ( 1st ` z ) )
3 fveq2 5887 . . . . . . . . . . . 12 |- ( x = z -> ( 2nd ` x ) = ( 2nd ` z ) )
42, 3uneq12d 3600 . . . . . . . . . . 11 |- ( x = z -> ( ( 1st ` x ) u. ( 2nd ` x ) ) = ( ( 1st ` z ) u. ( 2nd ` z ) ) )
5 eqid 2461 . . . . . . . . . . 11 |- ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) = ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) )
6 fvex 5897 . . . . . . . . . . . 12 |- ( 1st ` z ) e. _V
7 fvex 5897 . . . . . . . . . . . 12 |- ( 2nd ` z ) e. _V
86, 7unex 6615 . . . . . . . . . . 11 |- ( ( 1st ` z ) u. ( 2nd ` z ) ) e. _V
94, 5, 8fvmpt 5970 . . . . . . . . . 10 |- ( z e. ( On X. On ) -> ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` z ) = ( ( 1st ` z ) u. ( 2nd ` z ) ) )
10 fveq2 5887 . . . . . . . . . . . 12 |- ( x = w -> ( 1st ` x ) = ( 1st ` w ) )
11 fveq2 5887 . . . . . . . . . . . 12 |- ( x = w -> ( 2nd ` x ) = ( 2nd ` w ) )
1210, 11uneq12d 3600 . . . . . . . . . . 11 |- ( x = w -> ( ( 1st ` x ) u. ( 2nd ` x ) ) = ( ( 1st ` w ) u. ( 2nd ` w ) ) )
13 fvex 5897 . . . . . . . . . . . 12 |- ( 1st ` w ) e. _V
14 fvex 5897 . . . . . . . . . . . 12 |- ( 2nd ` w ) e. _V
1513, 14unex 6615 . . . . . . . . . . 11 |- ( ( 1st ` w ) u. ( 2nd ` w ) ) e. _V
1612, 5, 15fvmpt 5970 . . . . . . . . . 10 |- ( w e. ( On X. On ) -> ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` w ) = ( ( 1st ` w ) u. ( 2nd ` w ) ) )
179, 16breqan12d 4431 . . . . . . . . 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 4765 . . . . . . . . 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 269 . . . . . . . 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 2477 . . . . . . . . 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 716 . . . . . . . 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 721 . . . . . . 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 647 . . . . . 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 4480 . . . . 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 2486 . . . 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 6849 . . . . . . . 8 |- ( x e. ( On X. On ) -> ( 1st ` x ) e. On )
27 xp2nd 6850 . . . . . . . 8 |- ( x e. ( On X. On ) -> ( 2nd ` x ) e. On )
28 fvex 5897 . . . . . . . . . 10 |- ( 1st ` x ) e. _V
2928elon 5450 . . . . . . . . 9 |- ( ( 1st ` x ) e. On <-> Ord ( 1st ` x ) )
30 fvex 5897 . . . . . . . . . 10 |- ( 2nd ` x ) e. _V
3130elon 5450 . . . . . . . . 9 |- ( ( 2nd ` x ) e. On <-> Ord ( 2nd ` x ) )
32 ordun 5542 . . . . . . . . 9 |- ( ( Ord ( 1st ` x ) /\ Ord ( 2nd ` x ) ) -> Ord ( ( 1st ` x ) u. ( 2nd ` x ) ) )
3329, 31, 32syl2anb 486 . . . . . . . 8 |- ( ( ( 1st ` x ) e. On /\ ( 2nd ` x ) e. On ) -> Ord ( ( 1st ` x ) u. ( 2nd ` x ) ) )
3426, 27, 33syl2anc 671 . . . . . . 7 |- ( x e. ( On X. On ) -> Ord ( ( 1st ` x ) u. ( 2nd ` x ) ) )
3528, 30unex 6615 . . . . . . . 8 |- ( ( 1st ` x ) u. ( 2nd ` x ) ) e. _V
3635elon 5450 . . . . . . 7 |- ( ( ( 1st ` x ) u. ( 2nd ` x ) ) e. On <-> Ord ( ( 1st ` x ) u. ( 2nd ` x ) ) )
3734, 36sylibr 217 . . . . . 6 |- ( x e. ( On X. On ) -> ( ( 1st ` x ) u. ( 2nd ` x ) ) e. On )
385, 37fmpti 6067 . . . . 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 6636 . . . . 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 8467 . . . . 5 |- L We ( On X. On )
4443a1i 11 . . . 4 |- ( T. -> L We ( On X. On ) )
45 vex 3059 . . . . . . . 8 |- u e. _V
4645dmex 6752 . . . . . . 7 |- dom u e. _V
4745rnex 6753 . . . . . . 7 |- ran u e. _V
4846, 47unex 6615 . . . . . 6 |- ( dom u u. ran u ) e. _V
49 imadmres 5345 . . . . . . 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 3664 . . . . . . . . . 10 |- ( u i^i ( On X. On ) ) C_ ( On X. On )
51 ssun1 3608 . . . . . . . . . . . . . 14 |- dom u C_ ( dom u u. ran u )
5250sseli 3439 . . . . . . . . . . . . . . . . 17 |- ( x e. ( u i^i ( On X. On ) ) -> x e. ( On X. On ) )
53 1st2nd2 6856 . . . . . . . . . . . . . . . . 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 3663 . . . . . . . . . . . . . . . . 17 |- ( u i^i ( On X. On ) ) C_ u
5655sseli 3439 . . . . . . . . . . . . . . . 16 |- ( x e. ( u i^i ( On X. On ) ) -> x e. u )
5754, 56eqeltrrd 2540 . . . . . . . . . . . . . . 15 |- ( x e. ( u i^i ( On X. On ) ) -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. u )
5828, 30opeldm 5056 . . . . . . . . . . . . . . 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 3441 . . . . . . . . . . . . 13 |- ( x e. ( u i^i ( On X. On ) ) -> ( 1st ` x ) e. ( dom u u. ran u ) )
61 ssun2 3609 . . . . . . . . . . . . . 14 |- ran u C_ ( dom u u. ran u )
6228, 30opelrn 5084 . . . . . . . . . . . . . . 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 3441 . . . . . . . . . . . . 13 |- ( x e. ( u i^i ( On X. On ) ) -> ( 2nd ` x ) e. ( dom u u. ran u ) )
65 prssi 4140 . . . . . . . . . . . . 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 671 . . . . . . . . . . . 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 6679 . . . . . . . . . . . . 13 |- ( ( ( 1st ` x ) e. On /\ ( 2nd ` x ) e. On ) -> ( ( 1st ` x ) u. ( 2nd ` x ) ) e. { ( 1st ` x ) , ( 2nd ` x ) } )
7067, 68, 69syl2anc 671 . . . . . . . . . . . 12 |- ( x e. ( u i^i ( On X. On ) ) -> ( ( 1st ` x ) u. ( 2nd ` x ) ) e. { ( 1st ` x ) , ( 2nd ` x ) } )
7166, 70sseldd 3444 . . . . . . . . . . 11 |- ( x e. ( u i^i ( On X. On ) ) -> ( ( 1st ` x ) u. ( 2nd ` x ) ) e. ( dom u u. ran u ) )
7271rgen 2758 . . . . . . . . . 10 |- A. x e. ( u i^i ( On X. On ) ) ( ( 1st ` x ) u. ( 2nd ` x ) ) e. ( dom u u. ran u )
73 ssrab 3518 . . . . . . . . . 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 936 . . . . . . . . 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 5143 . . . . . . . . . 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 5756 . . . . . . . . . . 11 |- dom ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) = ( On X. On )
7776ineq2i 3642 . . . . . . . . . 10 |- ( u i^i dom ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ) = ( u i^i ( On X. On ) )
7875, 77eqtri 2483 . . . . . . . . 9 |- dom ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |` u ) = ( u i^i ( On X. On ) )
795mptpreima 5346 . . . . . . . . 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 3482 . . . . . . . 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 5636 . . . . . . . . 9 |- Fun ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) )
82 resss 5146 . . . . . . . . . 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 5052 . . . . . . . . . 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 6020 . . . . . . . . 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 683 . . . . . . . 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 214 . . . . . . 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 3474 . . . . . 6 |- ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " u ) C_ ( dom u u. ran u )
8948, 88ssexi 4561 . . . . 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 6938 . . 3 |- ( T. -> R We ( On X. On ) )
92 epse 4835 . . . . 5 |- _E Se On
9392a1i 11 . . . 4 |- ( T. -> _E Se On )
9445uniex 6613 . . . . . . . 8 |- U. u e. _V
9594pwex 4599 . . . . . . 7 |- ~P U. u e. _V
9695, 95xpex 6621 . . . . . 6 |- ( ~P U. u X. ~P U. u ) e. _V
975mptpreima 5346 . . . . . . . 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 2757 . . . . . . . 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 2483 . . . . . . 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 471 . . . . . . . . 9 |- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
101 elssuni 4240 . . . . . . . . . . . . 13 |- ( ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u -> ( ( 1st ` x ) u. ( 2nd ` x ) ) C_ U. u )
102101adantl 472 . . . . . . . . . . . 12 |- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> ( ( 1st ` x ) u. ( 2nd ` x ) ) C_ U. u )
103102unssad 3622 . . . . . . . . . . 11 |- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> ( 1st ` x ) C_ U. u )
10428elpw 3968 . . . . . . . . . . 11 |- ( ( 1st ` x ) e. ~P U. u <-> ( 1st ` x ) C_ U. u )
105103, 104sylibr 217 . . . . . . . . . 10 |- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> ( 1st ` x ) e. ~P U. u )
106102unssbd 3623 . . . . . . . . . . 11 |- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> ( 2nd ` x ) C_ U. u )
10730elpw 3968 . . . . . . . . . . 11 |- ( ( 2nd ` x ) e. ~P U. u <-> ( 2nd ` x ) C_ U. u )
108106, 107sylibr 217 . . . . . . . . . 10 |- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> ( 2nd ` x ) e. ~P U. u )
109105, 108jca 539 . . . . . . . . 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 6851 . . . . . . . . 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 675 . . . . . . . 8 |- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> x e. ( ~P U. u X. ~P U. u ) )
112111abssi 3515 . . . . . . 7 |- { x | ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) } C_ ( ~P U. u X. ~P U. u )
11399, 112eqsstri 3473 . . . . . 6 |- ( `' ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " u ) C_ ( ~P U. u X. ~P U. u )
11496, 113ssexi 4561 . . . . 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 6939 . . 3 |- ( T. -> R Se ( On X. On ) )
11791, 116jca 539 . 2 |- ( T. -> ( R We ( On X. On ) /\ R Se ( On X. On ) ) )
118117trud 1463 1 |- ( R We ( On X. On ) /\ R Se ( On X. On ) )
Colors of variables: wff setvar class
Syntax hints:   <-> wb 189   \/ wo 374   /\ wa 375   = wceq 1454   T. wtru 1455   e. wcel 1897   {cab 2447   A.wral 2748   {crab 2752   _Vcvv 3056   u. cun 3413   i^i cin 3414   C_ wss 3415   ~Pcpw 3962   {cpr 3981   <.cop 3985   U.cuni 4211   class class class wbr 4415   {copab 4473   |-> cmpt 4474   _E cep 4761   Se wse 4809   We wwe 4810   X. cxp 4850   `'ccnv 4851   dom cdm 4852   ran crn 4853   |` cres 4854   "cima 4855   Ord word 5440   Oncon0 5441   Fun wfun 5594   -->wf 5596   ` cfv 5600   1stc1st 6817   2ndc2nd 6818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-ord 5444  df-on 5445  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-1st 6819  df-2nd 6820
This theorem is referenced by:  infxpenlem  8469
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