Theorem br2ndeq 34029

Description: Uniqueness condition for the binary relation 2^nd. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)

Hypotheses

Ref	Expression
br1steq.1	⊢ 𝐴 ∈ V
br1steq.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
br2ndeq	⊢ (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵)

Proof of Theorem br2ndeq

Step	Hyp	Ref	Expression
1		br1steq.1	. 2 ⊢ 𝐴 ∈ V
2		br1steq.2	. 2 ⊢ 𝐵 ∈ V
3		br2ndeqg 7927	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵))
4	1, 2, 3	mp2an 690	1 ⊢ (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵)

Syntax hints:   ↔ wb 205   = wceq 1541   ∈ wcel 2106  Vcvv 3442  ⟨cop 4584   class class class wbr 5097  2^nd c2nd 7903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5248  ax-nul 5255  ax-pr 5377  ax-un 7655

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4275  df-if 4479  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-br 5098  df-opab 5160  df-mpt 5181  df-id 5523  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-fo 6490  df-fv 6492  df-2nd 7905

This theorem is referenced by:  dfrn5  34031  brtxp  34319  brpprod  34324  elfuns  34354  brimg  34376  brcup  34378  brcap  34379  brrestrict  34388