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Theorem op2ndd 5934
Description: Extract the second 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
op2ndd (𝐶 = ⟨𝐴, 𝐵⟩ → (2nd𝐶) = 𝐵)

Proof of Theorem op2ndd
StepHypRef Expression
1 fveq2 5318 . 2 (𝐶 = ⟨𝐴, 𝐵⟩ → (2nd𝐶) = (2nd ‘⟨𝐴, 𝐵⟩))
2 op1st.1 . . 3 𝐴 ∈ V
3 op1st.2 . . 3 𝐵 ∈ V
42, 3op2nd 5932 . 2 (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵
51, 4syl6eq 2137 1 (𝐶 = ⟨𝐴, 𝐵⟩ → (2nd𝐶) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1290  wcel 1439  Vcvv 2620  cop 3453  cfv 5028  2nd c2nd 5924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-iota 4993  df-fun 5030  df-fv 5036  df-2nd 5926
This theorem is referenced by:  xp2nd  5951  sbcopeq1a  5971  csbopeq1a  5972  eloprabi  5980  mpt2mptsx  5981  dmmpt2ssx  5983  fmpt2x  5984  fmpt2co  5995  df2nd2  5999  xporderlem  6010  xpf1o  6614  frecuzrdgtcl  9880  frecuzrdgfunlem  9887  fisumcom2  10893
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