Theorem br2ndeq 35966

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 7956	. 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 2113  Vcvv 3440  ⟨cop 4586   class class class wbr 5098  2^nd c2nd 7932

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-2nd 7934

This theorem is referenced by:  dfrn5  35968  brtxp  36072  brpprod  36077  elfuns  36107  brimg  36129  brcup  36131  brcap  36132  brrestrict  36143