Theorem 2ndrn 5809

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 103	. 2 |- ( ( Rel R /\ A e. R ) -> A e. R )
2		1st2nd 5807	. . 3 |- ( ( Rel R /\ A e. R ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
3	2, 1	eqeltrrd 2115	. 2 |- ( ( Rel R /\ A e. R ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. R )
4		1stexg 5794	. . . 4 |- ( A e. R -> ( 1st ` A ) e. _V )
5		2ndexg 5795	. . . 4 |- ( A e. R -> ( 2nd ` A ) e. _V )
6	4, 5	jca 290	. . 3 |- ( A e. R -> ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) )
7		opelrng 4566	. . . 4 |- ( ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V /\ <. ( 1st ` A ) , ( 2nd ` A ) >. e. R ) -> ( 2nd ` A ) e. ran R )
8	7	3expa 1104	. . 3 |- ( ( ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) /\ <. ( 1st ` A ) , ( 2nd ` A ) >. e. R ) -> ( 2nd ` A ) e. ran R )
9	6, 8	sylan 267	. 2 |- ( ( A e. R /\ <. ( 1st ` A ) , ( 2nd ` A ) >. e. R ) -> ( 2nd ` A ) e. ran R )
10	1, 3, 9	syl2anc 391	1 |- ( ( Rel R /\ A e. R ) -> ( 2nd ` A ) e. ran R )

Syntax hints:   -> wi 4   /\ wa 97   e. wcel 1393   _Vcvv 2557   <.cop 3378   ran crn 4346   Rel wrel 4350   ` cfv 4902   1stc1st 5765   2ndc2nd 5766

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910  df-1st 5767  df-2nd 5768

This theorem is referenced by: (None)