Theorem brsingle 35919

Description: The binary relation form of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Hypotheses

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

Assertion

Ref	Expression
brsingle	⊢ (𝐴Singleton𝐵 ↔ 𝐵 = {𝐴})

Proof of Theorem brsingle

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brsingle.1	. 2 ⊢ 𝐴 ∈ V
2		brsingle.2	. 2 ⊢ 𝐵 ∈ V
3		df-singleton 35864	. 2 ⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
4		brxp 5733	. . 3 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
5	1, 2, 4	mpbir2an 711	. 2 ⊢ 𝐴(V × V)𝐵
6		velsn 4641	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
7	1	ideq 5862	. . 3 ⊢ (𝑥 I 𝐴 ↔ 𝑥 = 𝐴)
8	6, 7	bitr4i 278	. 2 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 I 𝐴)
9	1, 2, 3, 5, 8	brtxpsd3 35898	1 ⊢ (𝐴Singleton𝐵 ↔ 𝐵 = {𝐴})

Syntax hints:   ↔ wb 206   = wceq 1539   ∈ wcel 2107  Vcvv 3479  {csn 4625   class class class wbr 5142   I cid 5576   × cxp 5682  Singletoncsingle 35840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-symdif 4252  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-eprel 5583  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568  df-1st 8015  df-2nd 8016  df-txp 35856  df-singleton 35864

This theorem is referenced by:  elsingles  35920  fnsingle  35921  fvsingle  35922  brapply  35940  brsuccf  35943  funpartlem  35944