Theorem op1std 6382

Description: Extract the first 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
op1std	|- ( C = <. A , B >. -> ( 1st ` C ) = A )

Proof of Theorem op1std

Step	Hyp	Ref	Expression
1		fveq2 5695	. 2 |- ( C = <. A , B >. -> ( 1st ` C ) = ( 1st ` <. A , B >. ) )
2		op1st.1	. . 3 |- A e. _V
3		op1st.2	. . 3 |- B e. _V
4	2, 3	op1st 6380	. 2 |- ( 1st ` <. A , B >. ) = A
5	1, 4	eqtrdi 2287	1 |- ( C = <. A , B >. -> ( 1st ` C ) = A )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   <.cop 3712   ` cfv 5377   1stc1st 6372

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fv 5385  df-1st 6374

This theorem is used by:  xp1st  6399  sbcopeq1a  6421  csbopeq1a  6422  eloprabi  6432  mpomptsx  6433  dmmpossx  6435  fmpox  6436  fmpoco  6452  df1st2  6455  xporderlem  6467  xpf1o  7144  mapunen  7151  fisumcom2  12205  fprodcom2fi  12393  txbas  15359  cnmpt1st  15389  txhmeo  15420  lgsquadlem1  16196  lgsquadlem2  16197