Theorem op2ndd 6356

Description: Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
op1st.1	|- A e. _V
op1st.2	|- B e. _V

Assertion

Ref	Expression
op2ndd	|- ( C = <. A , B >. -> ( 2nd ` C ) = B )

Proof of Theorem op2ndd

Step	Hyp	Ref	Expression
1		fveq2 5675	. 2 |- ( C = <. A , B >. -> ( 2nd ` C ) = ( 2nd ` <. A , B >. ) )
2		op1st.1	. . 3 |- A e. _V
3		op1st.2	. . 3 |- B e. _V
4	2, 3	op2nd 6354	. 2 |- ( 2nd ` <. A , B >. ) = B
5	1, 4	eqtrdi 2283	1 |- ( C = <. A , B >. -> ( 2nd ` C ) = B )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   _Vcvv 2815   <.cop 3697   ` cfv 5357   2ndc2nd 6346

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 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

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 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fv 5365  df-2nd 6348

This theorem is referenced by:  xp2nd  6373  sbcopeq1a  6394  csbopeq1a  6395  eloprabi  6405  mpomptsx  6406  dmmpossx  6408  fmpox  6409  fmpoco  6425  df2nd2  6429  xporderlem  6440  xpf1o  7110  mapunen  7117  frecuzrdgtcl  10798  frecuzrdgfunlem  10805  fisumcom2  12149  fprodcom2fi  12337  txbas  15249  cnmpt2nd  15280  txhmeo  15310