Theorem 2ndrn 6268

Description: The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.)

Assertion

Ref	Expression
2ndrn	|- ( ( Rel R /\ A e. R ) -> ( 2nd ` A ) e. ran R )

Proof of Theorem 2ndrn

Step	Hyp	Ref	Expression
1		simpr 110	. 2 |- ( ( Rel R /\ A e. R ) -> A e. R )
2		1st2nd 6266	. . 3 |- ( ( Rel R /\ A e. R ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
3	2, 1	eqeltrrd 2282	. 2 |- ( ( Rel R /\ A e. R ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. R )
4		1stexg 6252	. . . 4 |- ( A e. R -> ( 1st ` A ) e. _V )
5		2ndexg 6253	. . . 4 |- ( A e. R -> ( 2nd ` A ) e. _V )
6	4, 5	jca 306	. . 3 |- ( A e. R -> ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) )
7		opelrng 4909	. . . 4 |- ( ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V /\ <. ( 1st ` A ) , ( 2nd ` A ) >. e. R ) -> ( 2nd ` A ) e. ran R )
8	7	3expa 1205	. . 3 |- ( ( ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) /\ <. ( 1st ` A ) , ( 2nd ` A ) >. e. R ) -> ( 2nd ` A ) e. ran R )
9	6, 8	sylan 283	. 2 |- ( ( A e. R /\ <. ( 1st ` A ) , ( 2nd ` A ) >. e. R ) -> ( 2nd ` A ) e. ran R )
10	1, 3, 9	syl2anc 411	1 |- ( ( Rel R /\ A e. R ) -> ( 2nd ` A ) e. ran R )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2175   _Vcvv 2771   <.cop 3635   ran crn 4675   Rel wrel 4679   ` cfv 5270   1stc1st 6223   2ndc2nd 6224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-fo 5276  df-fv 5278  df-1st 6225  df-2nd 6226

This theorem is referenced by: (None)