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Theorem br2ndeqg 8035
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 8025 . . 3 ((𝐴𝑉𝐵𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
21eqeq1d 2736 . 2 ((𝐴𝑉𝐵𝑊) → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶𝐵 = 𝐶))
3 fo2nd 8033 . . . 4 2nd :V–onto→V
4 fofn 6822 . . . 4 (2nd :V–onto→V → 2nd Fn V)
53, 4ax-mp 5 . . 3 2nd Fn V
6 opex 5474 . . 3 𝐴, 𝐵⟩ ∈ V
7 fnbrfvb 6959 . . 3 ((2nd Fn V ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩2nd 𝐶))
85, 6, 7mp2an 692 . 2 ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩2nd 𝐶)
9 eqcom 2741 . 2 (𝐵 = 𝐶𝐶 = 𝐵)
102, 8, 93bitr3g 313 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  Vcvv 3477  cop 4636   class class class wbr 5147   Fn wfn 6557  ontowfo 6560  cfv 6562  2nd c2nd 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fo 6568  df-fv 6570  df-2nd 8013
This theorem is referenced by:  br2ndeq  35752  fv2ndcnv  35758  brxrn  38355
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