Theorem 2ndrn 7216

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		1st2nd 7214	. . 3 |- ( ( Rel R /\ A e. R ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2		simpr 477	. . 3 |- ( ( Rel R /\ A e. R ) -> A e. R )
3	1, 2	eqeltrrd 2702	. 2 |- ( ( Rel R /\ A e. R ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. R )
4		fvex 6201	. . 3 |- ( 1st ` A ) e. _V
5		fvex 6201	. . 3 |- ( 2nd ` A ) e. _V
6	4, 5	opelrn 5357	. 2 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. R -> ( 2nd ` A ) e. ran R )
7	3, 6	syl 17	1 |- ( ( Rel R /\ A e. R ) -> ( 2nd ` A ) e. ran R )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   <.cop 4183   ran crn 5115   Rel wrel 5119   ` cfv 5888   1stc1st 7166   2ndc2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  heicant  33444  mblfinlem1  33446