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Theorem op1std 7690
Description: Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
op1st.1 𝐴 ∈ V
op1st.2 𝐵 ∈ V
Assertion
Ref Expression
op1std (𝐶 = ⟨𝐴, 𝐵⟩ → (1st𝐶) = 𝐴)

Proof of Theorem op1std
StepHypRef Expression
1 fveq2 6667 . 2 (𝐶 = ⟨𝐴, 𝐵⟩ → (1st𝐶) = (1st ‘⟨𝐴, 𝐵⟩))
2 op1st.1 . . 3 𝐴 ∈ V
3 op1st.2 . . 3 𝐵 ∈ V
42, 3op1st 7688 . 2 (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
51, 4syl6eq 2877 1 (𝐶 = ⟨𝐴, 𝐵⟩ → (1st𝐶) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  Vcvv 3500  cop 4570  cfv 6352  1st c1st 7678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6312  df-fun 6354  df-fv 6360  df-1st 7680
This theorem is referenced by:  1st2val  7708  xp1st  7712  sbcopeq1a  7739  csbopeq1a  7740  eloprabi  7752  mpomptsx  7753  dmmpossx  7755  fmpox  7756  ovmptss  7779  fmpoco  7781  df1st2  7784  fsplit  7803  fsplitOLD  7804  frxp  7811  xporderlem  7812  fnwelem  7816  fimaproj  7820  xpf1o  8668  mapunen  8675  xpwdomg  9038  hsmexlem2  9838  fsumcom2  15119  fprodcom2  15328  qredeu  15992  isfuncd  17125  cofucl  17148  funcres2b  17157  funcpropd  17160  xpcco1st  17424  xpccatid  17428  1stf1  17432  2ndf1  17435  1stfcl  17437  prf1  17440  prfcl  17443  prf1st  17444  prf2nd  17445  evlf1  17460  evlfcl  17462  curf1fval  17464  curf11  17466  curf1cl  17468  curfcl  17472  hof1fval  17493  txbas  22091  cnmpt1st  22192  txhmeo  22327  ptuncnv  22331  ptunhmeo  22332  xpstopnlem1  22333  xkohmeo  22339  prdstmdd  22647  ucnimalem  22804  fmucndlem  22815  fsum2cn  23394  ovoliunlem1  24018  lgsquadlem1  25870  lgsquadlem2  25871  2sqreuop  25952  2sqreuopnn  25953  2sqreuoplt  25954  2sqreuopltb  25955  2sqreuopnnlt  25956  2sqreuopnnltb  25957  clwlkclwwlkfolem  27699  wlkl0  28060  eulerpartlemgs2  31524  hgt750lemb  31813  cvmliftlem15  32429  satfv1  32494  satfdmlem  32499  fmlasuc  32517  msubty  32658  msubco  32662  msubvrs  32691  filnetlem4  33613  finixpnum  34744  poimirlem4  34763  poimirlem15  34774  poimirlem20  34779  poimirlem26  34785  poimirlem28  34787  heicant  34794  dicelvalN  38181  rmxypairf1o  39373  unxpwdom3  39560  fgraphxp  39676  elcnvlem  39826  dvnprodlem2  42097  etransclem46  42431  ovnsubaddlem1  42718  dmmpossx2  44217  rrx2plordisom  44542
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