Theorem 2ndrn 6379

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 6377	. . 3 |- ( ( Rel R /\ A e. R ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
3	2, 1	eqeltrrd 2312	. 2 |- ( ( Rel R /\ A e. R ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. R )
4		1stexg 6363	. . . 4 |- ( A e. R -> ( 1st ` A ) e. _V )
5		2ndexg 6364	. . . 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 4991	. . . 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 2205   _Vcvv 2815   <.cop 3694   ran crn 4752   Rel wrel 4756   ` cfv 5354   1stc1st 6334   2ndc2nd 6335

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fo 5360  df-fv 5362  df-1st 6336  df-2nd 6337

This theorem is referenced by: (None)