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Theorem op1std 6791
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
StepHypRef Expression
1 fveq2 5864 . 2  |-  ( C  =  <. A ,  B >.  ->  ( 1st `  C
)  =  ( 1st `  <. A ,  B >. ) )
2 op1st.1 . . 3  |-  A  e. 
_V
3 op1st.2 . . 3  |-  B  e. 
_V
42, 3op1st 6789 . 2  |-  ( 1st `  <. A ,  B >. )  =  A
51, 4syl6eq 2524 1  |-  ( C  =  <. A ,  B >.  ->  ( 1st `  C
)  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3113   <.cop 4033   ` cfv 5586   1stc1st 6779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fv 5594  df-1st 6781
This theorem is referenced by:  1st2val  6807  xp1st  6811  sbcopeq1a  6833  csbopeq1a  6834  eloprabi  6843  mpt2mptsx  6844  dmmpt2ssx  6846  fmpt2x  6847  ovmptss  6861  fmpt2co  6863  df1st2  6866  fsplit  6885  frxp  6890  xporderlem  6891  fnwelem  6895  xpf1o  7676  mapunen  7683  xpwdomg  8007  hsmexlem2  8803  fsumcom2  13548  qredeu  14103  isfuncd  15088  cofucl  15111  funcres2b  15120  funcpropd  15123  xpcco1st  15307  xpccatid  15311  1stf1  15315  2ndf1  15318  1stfcl  15320  prf1  15323  prfcl  15326  prf1st  15327  prf2nd  15328  evlf1  15343  evlfcl  15345  curf1fval  15347  curf11  15349  curf1cl  15351  curfcl  15355  hof1fval  15376  txbas  19803  cnmpt1st  19904  txhmeo  20039  ptuncnv  20043  ptunhmeo  20044  xpstopnlem1  20045  xkohmeo  20051  prdstmdd  20357  ucnimalem  20518  fmucndlem  20529  fsum2cn  21110  ovoliunlem1  21648  lgsquadlem1  23357  lgsquadlem2  23358  usgrac  24027  edgss  24028  fimaproj  27499  eulerpartlemgs2  27959  cvmliftlem15  28383  fprodcom2  28691  finixpnum  29615  heicant  29626  filnetlem4  29802  rmxypairf1o  30451  unxpwdom3  30645  fgraphxp  30776  dmmpt2ssx2  31990  dicelvalN  35975
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