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Theorem br2ndeq 35276
Description: Uniqueness condition for the binary relation 2nd. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
Hypotheses
Ref Expression
br1steq.1 𝐴 ∈ V
br1steq.2 𝐵 ∈ V
Assertion
Ref Expression
br2ndeq (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵)

Proof of Theorem br2ndeq
StepHypRef Expression
1 br1steq.1 . 2 𝐴 ∈ V
2 br1steq.2 . 2 𝐵 ∈ V
3 br2ndeqg 7994 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵))
41, 2, 3mp2an 689 1 (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wcel 2098  Vcvv 3468  cop 4629   class class class wbr 5141  2nd c2nd 7970
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-2nd 7972
This theorem is referenced by:  dfrn5  35278  brtxp  35385  brpprod  35390  elfuns  35420  brimg  35442  brcup  35444  brcap  35445  brrestrict  35454
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