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Theorem br2ndeqg 7706
 Description: Uniqueness condition for the binary relation 2nd. (Contributed by Scott Fenton, 2-Jul-2020.) Revised to remove sethood hypothesis on 𝐶. (Revised by Peter Mazsa, 17-Jan-2022.)
Assertion
Ref Expression
br2ndeqg ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵))

Proof of Theorem br2ndeqg
StepHypRef Expression
1 op2ndg 7696 . . 3 ((𝐴𝑉𝐵𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
21eqeq1d 2826 . 2 ((𝐴𝑉𝐵𝑊) → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶𝐵 = 𝐶))
3 fo2nd 7704 . . . 4 2nd :V–onto→V
4 fofn 6588 . . . 4 (2nd :V–onto→V → 2nd Fn V)
53, 4ax-mp 5 . . 3 2nd Fn V
6 opex 5352 . . 3 𝐴, 𝐵⟩ ∈ V
7 fnbrfvb 6714 . . 3 ((2nd Fn V ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩2nd 𝐶))
85, 6, 7mp2an 688 . 2 ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩2nd 𝐶)
9 eqcom 2831 . 2 (𝐵 = 𝐶𝐶 = 𝐵)
102, 8, 93bitr3g 314 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1530   ∈ wcel 2106  Vcvv 3499  ⟨cop 4569   class class class wbr 5062   Fn wfn 6346  –onto→wfo 6349  ‘cfv 6351  2nd c2nd 7682 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 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fo 6357  df-fv 6359  df-2nd 7684 This theorem is referenced by:  br2ndeq  32900  fv2ndcnv  32906  brxrn  35494
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