Theorem ot3rdg 7184

Description: Extract the third member of an ordered triple. (See ot1stg 7182 comment.) (Contributed by NM, 3-Apr-2015.)

Assertion

Ref	Expression
ot3rdg	|- ( C e. V -> ( 2nd ` <. A , B , C >. ) = C )

Proof of Theorem ot3rdg

Step	Hyp	Ref	Expression
1		df-ot 4186	. . 3 |- <. A , B , C >. = <. <. A , B >. , C >.
2	1	fveq2i 6194	. 2 |- ( 2nd ` <. A , B , C >. ) = ( 2nd ` <. <. A , B >. , C >. )
3		opex 4932	. . 3 |- <. A , B >. e. _V
4		op2ndg 7181	. . 3 |- ( ( <. A , B >. e. _V /\ C e. V ) -> ( 2nd ` <. <. A , B >. , C >. ) = C )
5	3, 4	mpan 706	. 2 |- ( C e. V -> ( 2nd ` <. <. A , B >. , C >. ) = C )
6	2, 5	syl5eq 2668	1 |- ( C e. V -> ( 2nd ` <. A , B , C >. ) = C )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   <.cotp 4185   ` cfv 5888   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-ot 4186  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-2nd 7169

This theorem is referenced by:  oteqimp  7187  el2xptp0  7212  splval  13502  splcl  13503  ida2  16709  coa2  16719  mamufval  20191  msrval  31435  mapdhval  37013  hdmap1val  37088