Theorem op1std 5795

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 5198	. 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 5793	. 2 |- ( 1st ` <. A , B >. ) = A
5	1, 4	syl6eq 2129	1 |- ( C = <. A , B >. -> ( 1st ` C ) = A )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   _Vcvv 2601   <.cop 3401   ` cfv 4922   1stc1st 5785

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fv 4930  df-1st 5787

This theorem is referenced by:  xp1st  5812  sbcopeq1a  5833  csbopeq1a  5834  eloprabi  5842  mpt2mptsx  5843  dmmpt2ssx  5845  fmpt2x  5846  fmpt2co  5857  df1st2  5860  xporderlem  5872