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Theorem br2ndeqg 31377
Description: Uniqueness condition for binary relationship over the 2nd relationship. (Contributed by Scott Fenton, 2-Jul-2020.)
Assertion
Ref Expression
br2ndeqg ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵))

Proof of Theorem br2ndeqg
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 4370 . . . . . 6 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
21breq1d 4623 . . . . 5 (𝑥 = 𝐴 → (⟨𝑥, 𝑦⟩2nd 𝐶 ↔ ⟨𝐴, 𝑦⟩2nd 𝐶))
32bibi1d 333 . . . 4 (𝑥 = 𝐴 → ((⟨𝑥, 𝑦⟩2nd 𝐶𝐶 = 𝑦) ↔ (⟨𝐴, 𝑦⟩2nd 𝐶𝐶 = 𝑦)))
43imbi2d 330 . . 3 (𝑥 = 𝐴 → ((𝐶𝑋 → (⟨𝑥, 𝑦⟩2nd 𝐶𝐶 = 𝑦)) ↔ (𝐶𝑋 → (⟨𝐴, 𝑦⟩2nd 𝐶𝐶 = 𝑦))))
5 opeq2 4371 . . . . . 6 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
65breq1d 4623 . . . . 5 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩2nd 𝐶 ↔ ⟨𝐴, 𝐵⟩2nd 𝐶))
7 eqeq2 2632 . . . . 5 (𝑦 = 𝐵 → (𝐶 = 𝑦𝐶 = 𝐵))
86, 7bibi12d 335 . . . 4 (𝑦 = 𝐵 → ((⟨𝐴, 𝑦⟩2nd 𝐶𝐶 = 𝑦) ↔ (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵)))
98imbi2d 330 . . 3 (𝑦 = 𝐵 → ((𝐶𝑋 → (⟨𝐴, 𝑦⟩2nd 𝐶𝐶 = 𝑦)) ↔ (𝐶𝑋 → (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵))))
10 breq2 4617 . . . 4 (𝑧 = 𝐶 → (⟨𝑥, 𝑦⟩2nd 𝑧 ↔ ⟨𝑥, 𝑦⟩2nd 𝐶))
11 eqeq1 2625 . . . 4 (𝑧 = 𝐶 → (𝑧 = 𝑦𝐶 = 𝑦))
12 vex 3189 . . . . 5 𝑥 ∈ V
13 vex 3189 . . . . 5 𝑦 ∈ V
14 vex 3189 . . . . 5 𝑧 ∈ V
1512, 13, 14br2ndeq 31375 . . . 4 (⟨𝑥, 𝑦⟩2nd 𝑧𝑧 = 𝑦)
1610, 11, 15vtoclbg 3253 . . 3 (𝐶𝑋 → (⟨𝑥, 𝑦⟩2nd 𝐶𝐶 = 𝑦))
174, 9, 16vtocl2g 3256 . 2 ((𝐴𝑉𝐵𝑊) → (𝐶𝑋 → (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵)))
18173impia 1258 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036   = wceq 1480  wcel 1987  cop 4154   class class class wbr 4613  2nd c2nd 7112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855  df-2nd 7114
This theorem is referenced by:  fv2ndcnv  31383
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