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Theorem br2ndeqg 7338
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 7328 . . 3 ((𝐴𝑉𝐵𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
21eqeq1d 2773 . 2 ((𝐴𝑉𝐵𝑊) → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶𝐵 = 𝐶))
3 fo2nd 7336 . . . 4 2nd :V–onto→V
4 fofn 6258 . . . 4 (2nd :V–onto→V → 2nd Fn V)
53, 4ax-mp 5 . . 3 2nd Fn V
6 opex 5060 . . 3 𝐴, 𝐵⟩ ∈ V
7 fnbrfvb 6377 . . 3 ((2nd Fn V ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩2nd 𝐶))
85, 6, 7mp2an 672 . 2 ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩2nd 𝐶)
9 eqcom 2778 . 2 (𝐵 = 𝐶𝐶 = 𝐵)
102, 8, 93bitr3g 302 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cop 4322   class class class wbr 4786   Fn wfn 6026  ontowfo 6029  cfv 6031  2nd c2nd 7314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-2nd 7316
This theorem is referenced by:  br2ndeq  32009  fv2ndcnv  32017  brxrn  34478
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