Theorem 2ndrn 6368

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 6366	. . 3 |- ( ( Rel R /\ A e. R ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
3	2, 1	eqeltrrd 2310	. 2 |- ( ( Rel R /\ A e. R ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. R )
4		1stexg 6352	. . . 4 |- ( A e. R -> ( 1st ` A ) e. _V )
5		2ndexg 6353	. . . 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 4980	. . . 4 |- ( ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V /\ <. ( 1st ` A ) , ( 2nd ` A ) >. e. R ) -> ( 2nd ` A ) e. ran R )
8	7	3expa 1230	. . 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 2203   _Vcvv 2812   <.cop 3685   ran crn 4741   Rel wrel 4745   ` cfv 5343   1stc1st 6323   2ndc2nd 6324

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4221  ax-pow 4279  ax-pr 4314  ax-un 4545

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-un 3214  df-in 3216  df-ss 3223  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691  df-uni 3908  df-br 4103  df-opab 4165  df-mpt 4166  df-id 4405  df-xp 4746  df-rel 4747  df-cnv 4748  df-co 4749  df-dm 4750  df-rn 4751  df-iota 5303  df-fun 5345  df-fn 5346  df-f 5347  df-fo 5349  df-fv 5351  df-1st 6325  df-2nd 6326

This theorem is referenced by: (None)