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Theorem op1std 6014
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 5389 . 2 (𝐶 = ⟨𝐴, 𝐵⟩ → (1st𝐶) = (1st ‘⟨𝐴, 𝐵⟩))
2 op1st.1 . . 3 𝐴 ∈ V
3 op1st.2 . . 3 𝐵 ∈ V
42, 3op1st 6012 . 2 (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
51, 4syl6eq 2166 1 (𝐶 = ⟨𝐴, 𝐵⟩ → (1st𝐶) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wcel 1465  Vcvv 2660  cop 3500  cfv 5093  1st c1st 6004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fv 5101  df-1st 6006
This theorem is referenced by:  xp1st  6031  sbcopeq1a  6053  csbopeq1a  6054  eloprabi  6062  mpomptsx  6063  dmmpossx  6065  fmpox  6066  fmpoco  6081  df1st2  6084  xporderlem  6096  xpf1o  6706  fisumcom2  11175  txbas  12354  cnmpt1st  12384  txhmeo  12415
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