Theorem 2ndrn 6236

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 6234	. . 3 |- ( ( Rel R /\ A e. R ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
3	2, 1	eqeltrrd 2271	. 2 |- ( ( Rel R /\ A e. R ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. R )
4		1stexg 6220	. . . 4 |- ( A e. R -> ( 1st ` A ) e. _V )
5		2ndexg 6221	. . . 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 4894	. . . 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 2164   _Vcvv 2760   <.cop 3621   ran crn 4660   Rel wrel 4664   ` cfv 5254   1stc1st 6191   2ndc2nd 6192

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fo 5260  df-fv 5262  df-1st 6193  df-2nd 6194

This theorem is referenced by: (None)