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Theorem br2ndeq 30918
 Description: Uniqueness condition for binary relationship over the 2nd relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
Hypotheses
Ref Expression
br2ndeq.1 𝐴 ∈ V
br2ndeq.2 𝐵 ∈ V
br2ndeq.3 𝐶 ∈ V
Assertion
Ref Expression
br2ndeq (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵)

Proof of Theorem br2ndeq
StepHypRef Expression
1 br2ndeq.1 . . . 4 𝐴 ∈ V
2 br2ndeq.2 . . . 4 𝐵 ∈ V
31, 2op2nd 7068 . . 3 (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵
43eqeq1i 2615 . 2 ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶𝐵 = 𝐶)
5 fo2nd 7080 . . . 4 2nd :V–onto→V
6 fofn 6030 . . . 4 (2nd :V–onto→V → 2nd Fn V)
75, 6ax-mp 5 . . 3 2nd Fn V
8 opex 4859 . . 3 𝐴, 𝐵⟩ ∈ V
9 fnbrfvb 6146 . . 3 ((2nd Fn V ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩2nd 𝐶))
107, 8, 9mp2an 704 . 2 ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩2nd 𝐶)
11 eqcom 2617 . 2 (𝐵 = 𝐶𝐶 = 𝐵)
124, 10, 113bitr3i 289 1 (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   Fn wfn 5799  –onto→wfo 5802  ‘cfv 5804  2nd c2nd 7058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-2nd 7060 This theorem is referenced by:  br2ndeqg  30920  dfrn5  30922  brtxp  31157  brpprod  31162  elfuns  31192  brimg  31214  brcup  31216  brcap  31217  brrestrict  31226
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