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Theorem br2ndeq 33129
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 7698 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵))
41, 2, 3mp2an 691 1 (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wcel 2112  Vcvv 3444  cop 4534   class class class wbr 5033  2nd c2nd 7674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fo 6334  df-fv 6336  df-2nd 7676
This theorem is referenced by:  dfrn5  33131  brtxp  33455  brpprod  33460  elfuns  33490  brimg  33512  brcup  33514  brcap  33515  brrestrict  33524
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