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Mirrors  >  Home  >  MPE Home  >  Th. List  >  r0weon Structured version   Unicode version
Theorem r0weon 8391
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 5866 . . . . . . . . . . . 12 |- ( x = z -> ( 1st ` x ) = ( 1st ` z ) )
3 fveq2 5866 . . . . . . . . . . . 12 |- ( x = z -> ( 2nd ` x ) = ( 2nd ` z ) )
42, 3uneq12d 3659 . . . . . . . . . . 11 |- ( x = z -> ( ( 1st ` x ) u. ( 2nd ` x ) ) = ( ( 1st ` z ) u. ( 2nd ` z ) ) )
5 eqid 2467 . . . . . . . . . . 11 |- ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) = ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) )
6 fvex 5876 . . . . . . . . . . . 12 |- ( 1st ` z ) e. _V
7 fvex 5876 . . . . . . . . . . . 12 |- ( 2nd ` z ) e. _V
86, 7unex 6583 . . . . . . . . . . 11 |- ( ( 1st ` z ) u. ( 2nd ` z ) ) e. _V
94, 5, 8fvmpt 5951 . . . . . . . . . 10 |- ( z e. ( On X. On ) -> ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` z ) = ( ( 1st ` z ) u. ( 2nd ` z ) ) )
10 fveq2 5866 . . . . . . . . . . . 12 |- ( x = w -> ( 1st ` x ) = ( 1st ` w ) )
11 fveq2 5866 . . . . . . . . . . . 12 |- ( x = w -> ( 2nd ` x ) = ( 2nd ` w ) )
1210, 11uneq12d 3659 . . . . . . . . . . 11 |- ( x = w -> ( ( 1st ` x ) u. ( 2nd ` x ) ) = ( ( 1st ` w ) u. ( 2nd ` w ) ) )
13 fvex 5876 . . . . . . . . . . . 12 |- ( 1st ` w ) e. _V
14 fvex 5876 . . . . . . . . . . . 12 |- ( 2nd ` w ) e. _V
1513, 14unex 6583 . . . . . . . . . . 11 |- ( ( 1st ` w ) u. ( 2nd ` w ) ) e. _V
1612, 5, 15fvmpt 5951 . . . . . . . . . 10 |- ( w e. ( On X. On ) -> ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` w ) = ( ( 1st ` w ) u. ( 2nd ` w ) ) )
179, 16breqan12d 4462 . . . . . . . . 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 4793 . . . . . . . . 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 2490 . . . . . . . . 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 4511 . . . . 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 2499 . . . 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 6815 . . . . . . . 8 |- ( x e. ( On X. On ) -> ( 1st ` x ) e. On )
27 xp2nd 6816 . . . . . . . 8 |- ( x e. ( On X. On ) -> ( 2nd ` x ) e. On )
28 fvex 5876 . . . . . . . . . 10 |- ( 1st ` x ) e. _V
2928elon 4887 . . . . . . . . 9 |- ( ( 1st ` x ) e. On <-> Ord ( 1st ` x ) )
30 fvex 5876 . . . . . . . . . 10 |- ( 2nd ` x ) e. _V
3130elon 4887 . . . . . . . . 9 |- ( ( 2nd ` x ) e. On <-> Ord ( 2nd ` x ) )
32 ordun 4979 . . . . . . . . 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 6583 . . . . . . . 8 |- ( ( 1st ` x ) u. ( 2nd ` x ) ) e. _V
3635elon 4887 . . . . . . 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 6045 . . . . 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 6604 . . . . 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 8390 . . . . 5 |- L We ( On X. On )
4443a1i 11 . . . 4 |- ( T. -> L We ( On X. On ) )
45 vex 3116 . . . . . . . 8 |- u e. _V
4645dmex 6718 . . . . . . 7 |- dom u e. _V
4745rnex 6719 . . . . . . 7 |- ran u e. _V
4846, 47unex 6583 . . . . . 6 |- ( dom u u. ran u ) e. _V
49 imadmres 5499 . . . . . . 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 3719 . . . . . . . . . 10 |- ( u i^i ( On X. On ) ) C_ ( On X. On )
51 ssun1 3667 . . . . . . . . . . . . . 14 |- dom u C_ ( dom u u. ran u )
5250sseli 3500 . . . . . . . . . . . . . . . . 17 |- ( x e. ( u i^i ( On X. On ) ) -> x e. ( On X. On ) )
53 1st2nd2 6822 . . . . . . . . . . . . . . . . 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 3718 . . . . . . . . . . . . . . . . 17 |- ( u i^i ( On X. On ) ) C_ u
5655sseli 3500 . . . . . . . . . . . . . . . 16 |- ( x e. ( u i^i ( On X. On ) ) -> x e. u )
5754, 56eqeltrrd 2556 . . . . . . . . . . . . . . 15 |- ( x e. ( u i^i ( On X. On ) ) -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. u )
5828, 30opeldm 5206 . . . . . . . . . . . . . . 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 3502 . . . . . . . . . . . . 13 |- ( x e. ( u i^i ( On X. On ) ) -> ( 1st ` x ) e. ( dom u u. ran u ) )
61 ssun2 3668 . . . . . . . . . . . . . 14 |- ran u C_ ( dom u u. ran u )
6228, 30opelrn 5234 . . . . . . . . . . . . . . 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 3502 . . . . . . . . . . . . 13 |- ( x e. ( u i^i ( On X. On ) ) -> ( 2nd ` x ) e. ( dom u u. ran u ) )
65 prssi 4183 . . . . . . . . . . . . 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 6646 . . . . . . . . . . . . 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 3505 . . . . . . . . . . 11 |- ( x e. ( u i^i ( On X. On ) ) -> ( ( 1st ` x ) u. ( 2nd ` x ) ) e. ( dom u u. ran u ) )
7271rgen 2824 . . . . . . . . . 10 |- A. x e. ( u i^i ( On X. On ) ) ( ( 1st ` x ) u. ( 2nd ` x ) ) e. ( dom u u. ran u )
73 ssrab 3578 . . . . . . . . . 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 918 . . . . . . . . 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 5294 . . . . . . . . . 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 5736 . . . . . . . . . . 11 |- dom ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) = ( On X. On )
7776ineq2i 3697 . . . . . . . . . 10 |- ( u i^i dom ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ) = ( u i^i ( On X. On ) )
7875, 77eqtri 2496 . . . . . . . . 9 |- dom ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |` u ) = ( u i^i ( On X. On ) )
795mptpreima 5500 . . . . . . . . 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 3543 . . . . . . . 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 5624 . . . . . . . . 9 |- Fun ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) )
82 resss 5297 . . . . . . . . . 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 5202 . . . . . . . . . 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 5998 . . . . . . . . 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 3535 . . . . . 6 |- ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " u ) C_ ( dom u u. ran u )
8948, 88ssexi 4592 . . . . 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 6900 . . 3 |- ( T. -> R We ( On X. On ) )
92 epse 4862 . . . . 5 |- _E Se On
9392a1i 11 . . . 4 |- ( T. -> _E Se On )
9445uniex 6581 . . . . . . . 8 |- U. u e. _V
9594pwex 4630 . . . . . . 7 |- ~P U. u e. _V
9695, 95xpex 6589 . . . . . 6 |- ( ~P U. u X. ~P U. u ) e. _V
975mptpreima 5500 . . . . . . . 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 2823 . . . . . . . 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 2496 . . . . . . 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 4275 . . . . . . . . . . . . 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 3681 . . . . . . . . . . 11 |- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> ( 1st ` x ) C_ U. u )
10428elpw 4016 . . . . . . . . . . 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 3682 . . . . . . . . . . 11 |- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> ( 2nd ` x ) C_ U. u )
10730elpw 4016 . . . . . . . . . . 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 6817 . . . . . . . . 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 3575 . . . . . . 7 |- { x | ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) } C_ ( ~P U. u X. ~P U. u )
11399, 112eqsstri 3534 . . . . . 6 |- ( `' ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " u ) C_ ( ~P U. u X. ~P U. u )
11496, 113ssexi 4592 . . . . 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 6901 . . 3 |- ( T. -> R Se ( On X. On ) )
11791, 116jca 532 . 2 |- ( T. -> ( R We ( On X. On ) /\ R Se ( On X. On ) ) )
118117trud 1388 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 1379   T. wtru 1380   e. wcel 1767   {cab 2452   A.wral 2814   {crab 2818   _Vcvv 3113   u. cun 3474   i^i cin 3475   C_ wss 3476   ~Pcpw 4010   {cpr 4029   <.cop 4033   U.cuni 4245   class class class wbr 4447   {copab 4504   |-> cmpt 4505   _E cep 4789   Se wse 4836   We wwe 4837   Ord word 4877   Oncon0 4878   X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000   |` cres 5001   "cima 5002   Fun wfun 5582   -->wf 5584   ` cfv 5588   1stc1st 6783   2ndc2nd 6784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-1st 6785  df-2nd 6786
This theorem is referenced by:  infxpenlem  8392
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