Theorem br2ndeq 35735

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 8009	. 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 1540   ∈ wcel 2108  Vcvv 3459  ⟨cop 4607   class class class wbr 5119  2^nd c2nd 7985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-fo 6536  df-fv 6538  df-2nd 7987

This theorem is referenced by:  dfrn5  35737  brtxp  35844  brpprod  35849  elfuns  35879  brimg  35901  brcup  35903  brcap  35904  brrestrict  35913