Theorem br2ndeq 35808

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 7939	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵))
4	1, 2, 3	mp2an 692	1 ⊢ (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵)

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⟨cop 4577   class class class wbr 5086  2^nd c2nd 7915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fo 6482  df-fv 6484  df-2nd 7917

This theorem is referenced by:  dfrn5  35810  brtxp  35914  brpprod  35919  elfuns  35949  brimg  35971  brcup  35973  brcap  35974  brrestrict  35983